L∞-algebras and their cohomology

Ben Reinhold

doi:10.1051/emsci/2019003

Open Access

Issue		Emergent Scientist Volume 3, 2019


Article Number		4
Number of page(s)		17
Section		Mathematics
DOI		https://doi.org/10.1051/emsci/2019003
Published online		04 June 2019

Emergent Scientist 3, 4 (2019)

Research Article

L_∞-algebras and their cohomology

Ben Reinhold^*

University of Göttingen Mathematical Institute, 37073 Göttingen, Germany

^* e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 29 January 2019
Accepted: 23 April 2019

Abstract

We give an overview of different characterisations of L_∞-structures in terms of symmetric brackets and (co)differentials on the symmetric (co)algebra. We then do the same for their representations (up to homotopy) and approach L_∞-algebra cohomology using the commutator bracket on the space of coderivations of the symmetric coalgebra. This leads to abelian extensions of L_∞-algebras by 2-cocycles.

Key words: L_∞-algebras / representations (up to homotopy) / L_∞-algebra cohomology / abelian extensions by 2-cocycles

© B. Reinhold, published by EDP Sciences, 2019

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1 Introduction

L_∞-algebras (also called strongly homotopy Lie algebras) were first introduced in [1] and [2] and are a generalisation of graded Lie algebras in which a system of antisymmetric n-ary brackets satisfies a generalised Jacobi identity. The first part of this article serves as a self-contained introduction to L_∞-algebras, in which we discuss different characterisations of L_∞-algebras and their representations (up to homotopy), closely following [3].

The L_∞-algebra cohomology with values in the adjoint representation was introduced in [4] using a Lie bracket on the space of cochains. We extend this approach to arbitrary representations, which leads to a characterisation of certain L_∞-algebras as abelian extensions of L_∞-algebras by 2-cocycles. This generalises a theorem from [5] that characterises certain L_∞-algebras in terms of Lie algebra cohomology.

This article is largely based on my same-titled Bachelor’s thesis, which I wrote under the supervision of Chenchang Zhu at the University of Göttingen in 2018.

2 Mathematical background

In this section, we discuss exterior and symmetric powers, algebras and coalgebras in the graded framework. In particular, we show that antisymmetric and symmetric maps are related by a shift in degree and that coderivations of the symmetric coalgebra are in one-to-one correspondence with their weight one components. These results are later key to the characterisations of L_∞-structures in terms of symmetric brackets and codifferentials. The main references for this section are [3,4,6,7].

2.1 Graded vector spaces

A graded vector space is a vector space V together with a decomposition $V ≅ \underset{p \in ℤ}{\oplus} V_{p}$ $Mathematical equation: $V \cong \bigoplus_{p \in\Z}V_p$$ for a family of vector spaces ${V_{p}}_{p \in ℤ}$ $Mathematical equation: $\{V_p\}_{p\in\Z}$$ . An element v ∈ V _p is then called homogeneous of degree p and we write |v| = p.

Here and subsequently, we assume all vector spaces to be over a fixed ground field $k$ $Mathematical equation: ${\mathbb k}$$ of characteristic zero. We always denote by V and W graded vector spaces and by v₁, …, v_n ∈ V arbitrary homogeneous elements.

A linear map f : V → W is called homogeneous (of degree p) if there is $p \in ℤ$ $Mathematical equation: $p\in \Z$$ such that f(V _n) ⊂ W_n+p for all $n \in ℤ$ $Mathematical equation: $n \in \Z$$ . We denote by Hom _p(V, W) the vector space of all homogeneous linear maps V → W of degree p and by Hom (V, W) the graded vector space $\underset{p \in ℤ}{\oplus} {Hom}_{p} (V, W)$ $Mathematical equation: $\bigoplus_{p\in \Z}\Hom_p(V,W)$$ . Elements in Hom₀(V, W) are also called degree preserving.

Note that we can identify ungraded vector spaces with graded ones that are concentrated in degree zero, that is V _k = 0 for k≠ 0.

There is a canonical grading on the direct sum of V and W given by ${(V \oplus W)}_{p} = V_{p} \oplus W_{p}$ $Mathematical equation: $(V\oplus W)_p=V_p\oplus W_p$$ . The isomorphism $V \otimes W ≅ \underset{p \in ℤ}{\oplus} \underset{i + j = p}{\oplus} (V_{i} \otimes W_{j})$ $Mathematical equation: \[ V\otimes W \cong \bigoplus_{p \in \Z} \bigoplus_{i+j=p}\left(V_i\otimes W_j\right) \]$

allows us to define a grading on V ⊗ W by ${(V \otimes W)}_{p} = \underset{i + j = p}{\oplus} (V_{i} \otimes W_{j}) .$ $Mathematical equation: \[\left(V\otimes W\right)_p = \bigoplus_{i+j=p}\left(V_i\otimes W_j\right). \]$

This extends to a grading on $V^{\otimes n} ≔ \otimes_{i = 1}^{n} V$ $Mathematical equation: ${V^{\otimes n}} \vcentcolon=\bigotimes_{i=1}^nV$$ given by ${(V^{\otimes n})}_{p} = \underset{i_{1} + … + i_{n} = p}{\oplus} V_{i_{1}} \otimes … \otimes V_{i_{n}} .$ $Mathematical equation: \[({V^{\otimes n}})_p=\bigoplus_{i_1+\dotsc +i_n=p}V_{i_1}\otimes \dotsc \otimes V_{i_n}. \]$

We denote by τ_V,W the linear degree preserving map $τ_{V, W} : V \otimes W \to W \otimes V, v \otimes w \mapsto {(- 1)}^{| v | | w |} w \otimes v .$ $Mathematical equation: \[ \tau_{V,W}\colon V\otimes W\to W\otimes V,\quad v\otimes w\mapsto (-1)^{|v||w|}w\otimes v. \]$

If f ∈ Hom(V, W) and g ∈ Hom(V ′, W′) are homogeneous for graded vector spaces V ′ and W′, we define the linear map f ⊗ g: V ⊗ V ′→ W ⊗ W′ by $(f \otimes g) (v \otimes v^{'}) = {(- 1)}^{| v | | g |} f (v) \otimes g (v^{'})$ $Mathematical equation: \begin{equation*}(f\otimes g) (v\otimes v')= (-1)^{|v| |g|}f(v)\otimes g(v') \end{equation*}$ (1)

for v ∈ V , v′ ∈ V ′ homogeneous. Note that |f ⊗ g| = |f| + |g|. This generalises to tensor products of three or more vector spaces in the obvious way and we abbreviate f ⊗ … ⊗ f : V ^⊗n → W^⊗n to f^⊗n .

For the composition of such functions, (1) implies $(f^{'} \otimes g^{'}) ○ (f \otimes g) = {(- 1)}^{| g^{'} | | f |} (f^{'} ○ f) \otimes (g^{'} ○ g),$ $Mathematical equation: \begin{equation*}(f' \otimes g')\circ (f\otimes g)= (-1)^{|g'||f|}(f'\circ f)\otimes (g'\circ g), \end{equation*}$ (2)

when f′ and g′ are homogeneous linear maps with domains W and W′, respectively.

When working in the framework of graded vector spaces, the general rule of thumb for the signs is that whenever two “graded symbols” of degree p and q, respectively, change their order in an equation, there should be the sign ${(- 1)}^{| p | | q |}$ $Mathematical equation: $(-1)^{|p||q|}$$ . This is called the Koszul sign convention.

We denote by 𝔖_n the symmetric group, the group of all permutations of the set {1, …, n}, and by s_i ∈ 𝔖_n for 1 ≤ i≤ n − 1 the transposition with s_i(i) = i + 1 and s_i (i + 1) = i. There are two canonical linear right actions of 𝔖_n on V ^⊗n. These are given on the generating subset {s₁, …s_n−1}⊂ 𝔖_n by $\begin{array}{l} \hat{ε} (s_{i}) (v_{1} \otimes … \otimes v_{n}) & = {(- 1)}^{| v_{i} | | v_{i + 1} |} v_{1} \otimes … \otimes v_{i - 1} \\ \otimes v_{i + 1} \otimes v_{i} \otimes v_{i + 2} \otimes … \otimes v_{n}, \\ \hat{χ} (s_{i}) (v_{1} \otimes … \otimes v_{n}) & = - {(- 1)}^{| v_{i} | | v_{i + 1} |} v_{1} \otimes … \otimes v_{i - 1} \\ \otimes v_{i + 1} \otimes v_{i} \otimes v_{i + 2} \otimes … \otimes v_{n} . \end{array}$ $Mathematical equation: \begin{align*} \hat{\varepsilon}(s_i)(v_1\otimes\dotsc\otimes v_n) &=(-1)^{|v_i||v_{i+1}|}v_1\otimes \dotsc \otimes v_{i-1}\\ &\quad\otimes v_{i+1} \otimes v_i \otimes v_{i+2}\otimes \dotsc \otimes v_n,\\ \hat{\chi}( s_i)(v_1\otimes\dotsc\otimes v_n) &= -(-1)^{|v_i||v_{i+1}|}v_1\otimes \dotsc \otimes v_{i-1}\\ &\quad\otimes v_{i+1} \otimes v_i \otimes v_{i+2}\otimes \dotsc \otimes v_n. \end{align*}$

We call $\hat{ε}$ $Mathematical equation: $\hatsilon$$ and $\hat{χ}$ $Mathematical equation: $\hat\chi$$ the (graded) symmetric and (graded) antisymmetric action of 𝔖_n on V ^⊗n, respectively. Note that $\hat{ε} (σ)$ $Mathematical equation: $\hatsilon(\sigma)$$ is degree preserving for all σ ∈ 𝔖_n as $\hat{ε} (σ) (v_{1} \otimes … \otimes v_{n}) = \pm v_{σ (1)} \otimes … \otimes v_{σ (n)} .$ $Mathematical equation: \begin{equation*}\hat\varepsilon(\sigma)(v_1\otimes\dotsc\otimes v_n)=\pm v_{\sigma(1)}\otimes\dotsc\otimes v_{\sigma(n)}. \end{equation*}$ (3)

We then denote the sign in (3) by ε(σ;v₁, …, v_n) and similarly by χ(σ;v₁, …, v_n) the sign such that $\hat{χ} (σ) (v_{1} \otimes … \otimes v_{n}) = χ (σ; v_{1}, \dots, v_{n}) v_{σ (1)} \otimes … \otimes v_{σ (n)} .$ $Mathematical equation: \[\hat\chi(\sigma)(v_1\otimes\dotsc\otimes v_n)=\chi(\sigma;v_1,\dots,v_n)v_{\sigma(1)}\otimes\dotsc\otimes v_{\sigma(n)}.\]$

We abbreviate ε(σ;v₁, …, v_n) and χ(σ;v₁, …, v_n) to ε(σ) and χ(σ), when no confusion can arise.

Let U_S ⊂ V ^⊗n be the graded subspace spanned by all elements of the form $v_{1} \otimes … \otimes v_{n} - \hat{ε} (σ) (v_{1} \otimes … \otimes v_{n})$ $Mathematical equation: \[ v_1\otimes\dotsc\otimes v_n-\hatsilon(\sigma)(v_1\otimes\dotsc\otimes v_n) \]$

for σ ∈ 𝔖_n. The space $S^{n} (V) ≔ V^{\otimes n} / U_{S}$ $Mathematical equation: $\Sy^n(V)\vcentcolon= {V^{\otimes n}}/U_S$$ is called the nth symmetric power of V . Similarly, the nth exterior power of V is defined as the quotient of V ^⊗n by the graded subspace spanned by all elements of the form $v_{1} \otimes … \otimes v_{n} - \hat{χ} (σ) (v_{1} \otimes … \otimes v_{n})$ $Mathematical equation: \[ v_1\otimes\dotsc\otimes v_n-\hat\chi(\sigma)(v_1\otimes\dotsc\otimes v_n) \]$

for σ ∈ 𝔖_n and is denoted by ∧ⁿV.

An n-linear map f : V ⁿ → W is called (graded) symmetric iffor all σ ∈ 𝔖_n $f (v_{1}, …, v_{n}) = ε (σ) f (v_{σ (1)}, …, v_{(n)})$ $Mathematical equation: \[ f(v_1,\dotsc,v_n)=\varepsilon(\sigma)f(v_{\sigma(1)},\dotsc, v_{(n)}) \]$

holds. We can write this conveniently as $f ○ \hat{ε} (σ) = f$ $Mathematical equation: $f\circ \hatsilon(\sigma)=f$$ . Similarly, f is called (graded) antisymmetric if $f ○ \hat{χ} (σ) = f$ $Mathematical equation: $f\circ \hat\chi(\sigma)=f$$ for all σ ∈ 𝔖_n.

Proposition 1. Let f : V ^⊗n → W be a symmetric linear map. There is a unique linear map $φ : S^{n} (V) \to W$ $Mathematical equation: $\varphi\colon\Sy^n(V)\to W$$ such that the following diagram commutes: where $π_{S} : V^{\otimes n} \to S^{n} (V)$ $Mathematical equation: $\pi_S\colon {V^{\otimes n}} \to \Sy^n(V)$$ is the canonical projection.

Proof. As f is symmetric, it vanishes on the generators of U_S and factors through π_S to a linear map $φ : S^{n} (V) \to W$ $Mathematical equation: $\varphi\colon \Sy^n(V)\to W$$ such that the diagram above commutes. This map is unique as π_S is surjective.

Remark 2. As the symmetric 𝔖_n-action on V^⊗n is degree preserving, $S^{n} (V)$ $Mathematical equation: $\Sy^n(V)$$ inherits a canonical grading from V ^⊗n such that π_S is degree preserving. It is then immediate that if f is homogeneous in Proposition 1, so is the map φ and |φ| = |f|. As π_S is symmetricby construction of $S^{n} (V)$ $Mathematical equation: $\Sy^n(V)$$ , Proposition 1 yields an isomorphism between the subspace of Hom(V ^⊗n, W) consisting of all symmetric maps and $Hom (S^{n} (V), W)$ $Mathematical equation: $\Hom(\Sy^n(V),W)$$ . An analogue of Proposition 1 holds for ∧ ⁿV and induces an isomorphism between the subspace Hom(V ^⊗n, W) of all anti-symmetric maps and Hom(∧ ⁿV, W).

An element in V ^⊗n is called symmetric if it is invariant under the symmetric 𝔖_n-action on V ^⊗n. We claim that $S^{n} (V)$ $Mathematical equation: $\Sy^n(V)$$ is isomorphic to the subspace of V ^⊗n of all symmetric elements. Indeed, letting v₁ ∨… ∨ v_n denote the image of v₁ ⊗… ⊗ v_n under π_S, the linear map $\begin{array}{l} φ : S^{n} (V) \to V^{\otimes n}, \\ x_{1} \lor … \lor x_{n} \mapsto \frac{1}{n!} \sum_{σ \in S_{n}} ε (σ) x_{σ (1)} \otimes … \otimes x_{σ (n)} \end{array}$ $Mathematical equation: \[ \begin{array}{l} \varphi\colon \Sy^n(V)\to {V^{\otimes n}}, \\ x_1\vee\dotsc\vee x_n\mapsto \frac{1}{n!}\sum_{\sigma\in \mathfrak S_n}\varepsilon(\sigma)x_{\sigma(1)}\otimes\dotsc\otimes x_{\sigma(n)} \end{array} \]$

is well-defined and satisfies $π_{S} ○ φ = {id}_{S^{n} (V)}$ $Mathematical equation: $\pi_S\circ \varphi=\id_{\Sy^n(V)}$$ and $φ ○ π_{S} = \frac{1}{n!} \sum_{σ \in S_{n}} \hat{ε} (σ)$ $Mathematical equation: $\varphi\circ \pi_S=\frac{1}{n!}\sum_{\sigma\in\mathfrak S_n}\hatsilon(\sigma)$$ . As the latter is a projection of V ^⊗n onto said subspace, the claim follows. A similar statement clearly holds for ∧ⁿV .

For $n \in ℤ$ $Mathematical equation: $n\in\Z$$ , we defProposition2 ine the graded vector space V [n] to be the vector space V with the grading defined by $V {[n]}_{p} = V_{p + n}$ $Mathematical equation: $V[n]_p= V_{p+n}$$ . We denote by ↓ⁿ: V → V [n] the identity map on V , which becomes a linear isomorphism of degree − n, and by ↑ⁿ its inverse. We abbreviate ↓¹ and ↑¹ to ↓ and ↑, respectively. Note that ${(↓^{\otimes k})}^{- 1} = {(- 1)}^{\frac{k (k - 1)}{2}} ↑^{\otimes k}$ $Mathematical equation: $(\downarrow^{\otimes k})^{-1}=(-1)^{\frac{k(k-1)}{2}}\uparrow^{\otimes k}$$ as a consequence of (2).

Proposition 3 (The décalage isomorphism). For σ ∈ 𝔖_n,

$\hat{ε} (σ) ○ ↓^{\otimes n} = ↓^{\otimes n} ○ \hat{χ} (σ) .$ $Mathematical equation: \begin{equation*}\hat\varepsilon(\sigma)\circ \downarrow^{\otimes n} = \downarrow^{\otimes n} \circ \hat\chi(\sigma). \end{equation*}$ (4)

There is then a degree preserving isomorphism $S^{n} (V [1]) ≅ (\Asy^{n} V) [n] .$ $Mathematical equation: \begin{equation*}\Sy^n(V[1])\cong\left(\Asy^n V\right)[n]. \end{equation*}$ (5)

Proof. Note that for the first part, we only have to check (4) on the generating subset {s₁ , …, s_n−1}⊂ 𝔖_n. This is an easy computation left to the reader. Let π_A: V ^⊗n →∧ⁿV be the canonical projection. The linear maps $\begin{array}{@{}c@{}c@{}l@{}}nn& π_{S} ○ ↓ &^\otimes n : V^{\otimes n} \to S (V [1]), {(- 1)}^{\frac{n (n - 1)}{2}} & π_{A} ○ ↑ &^\otimes n : {(V [1])}^{\otimes n} \to \Asy^{n} V nn\end{array}$ $Mathematical equation: \[ \begin{array}{@{}c@{}c@{}l@{}} &\pi_S\circ \downarrow&^{\otimes n}\colon {V^{\otimes n}} \to \Sy(V[1]),\\ (-1)^{\frac{n(n-1)}{2}}&\pi_A\circ\uparrow&^{\otimes n}\colon (V[1])^{\otimes n}\to \Asy^n V \end{array} \]$

are then antisymmetric and symmetric, respectively. The induced linear maps between $S (V [1])$ $Mathematical equation: $\Sy(V[1])$$ and ∧ⁿV are then easily seen to be inverse to each other. As these maps are of degree − n and n, respectively, we obtain a degree preserving isomorphism $S (V [1]) ≅ (\Asy^{n} V) [n]$ $Mathematical equation: $\Sy(V[1])\cong \big(\Asy^n V\big)[n]$$ .

Corollary 1. There is for each $p \in ℤ$ $Mathematical equation: $p\in\Z$$ a one-to-one correspondence between symmetric linear maps $λ : {(V [1])}^{\otimes n} \to V [1]$ $Mathematical equation: $\lambda\colon (V[1])^{\otimes n}\to V[1]$$ of degree p and antisymmetric linear maps l: V ^⊗n → V of degree p + 1 − n given by

$\begin{matrix} l & = ↑ ○ λ ○ ↓^{\otimes n}, \\ λ & = {(- 1)}^{\frac{n (n - 1)}{2}} ↓ ○ l ○ ↑^{\otimes n} . \end{matrix}$ $Mathematical equation: \[ \begin{array}{@{}l@{}l@{}} l&=\uparrow\circ\lambda \circ \downarrow^{\otimes n},\\ \lambda&=(-1)^{\frac{n(n-1)}{2}}\downarrow\circ l\circ \uparrow^{\otimes n}. \end{array} \]$

A differential on the graded vector space V is a linear map d: V → V of degree one such that d² = 0. We then callthe pair (V, d) a differential graded vector space (DG vector space for short). A homomorphism between DG vector spaces (V, d) and (W, d′) is a degree preserving linear map f : V → W such that d′ ○ f = f ○ d.

DG vector spaces are sometimes called cochain complexes. Given a cochain complex (V, d), one then calls an element v ∈ V _n an n-cocycle if d(v) = 0 and an n-coboundary if v = d(w) for some w ∈ V _n−1. The graded vector space H(V ) = ker(d)∕im(d) measures the non-exactness of the sequence $\dots \overset{d}{\to} V_{n - 1} \overset{d}{\to} V_{n} \overset{d}{\to} V_{n + 1} \overset{d}{\to} \dots$ $Mathematical equation: \[\cdots\xrightarrow{d} V_{n-1} \xrightarrow{d} V_n \xrightarrow{d}V_{n+1} \xrightarrow{d}\cdots\]$

and is called the cohomology of (V, d). We then call $H_{n} (V) = \frac{{$ $Mathematical equation: $H_n(V)= \frac{\text{$$ the nth cohomology group.

2.2 Graded algebras

By an algebra we mean a vector space A together with a linear map μ: A ⊗ A → A; the multiplication μ is in generalnot assumed to be associative.

A graded algebra A is an algebra that is also a graded vector space in which the multiplication is degree preserving. If also $a b = {(- 1)}^{| a | | b |} b a$ $Mathematical equation: $a b=(-1)^{|a||b|}b a$$ for all a, b ∈ A, we call A (graded) commutative. A homomorphism of graded algebras is a degree preserving algebra homomorphism.

A (two-sided) ideal I in A is called homogeneous if I ⊂ A is a graded subspace. Note that an ideal is homogeneous if and only if it is spanned by homogeneous elements.

Remark 5. If I ⊂ A is a homogeneous ideal, the canonical isomorphism $A / I ≅ \oplus_{n \in ℤ} A_{n} / I_{n}$ $Mathematical equation: $A/I\cong \bigoplus_{n\in \Z} A_n/I_n$$ makes A∕I into a graded algebra such that the canonical projection A → A∕I is a homomorphism of graded algebras.

Remark 6. Let A and B be two graded associative algebras. The multiplication defined by $(a \otimes b) (a^{'} \otimes b^{'}) = {(- 1)}^{| b | | a^{'} |} a a^{'} \otimes b b^{'} .$ $Mathematical equation: \begin{equation*}(a\otimes b)(a'\otimes b')= (-1)^{|b||a'|}a a'\otimes b b'. \end{equation*}$ (6) for a, a′∈ A, b, b′ ∈ B homogeneous makes A ⊗ B into a graded associative algebra. If A and B are both unital/commutative, then so is A ⊗ B.

Example 7 (The tensor algebra). We denote by T(V ) := ⊕ _n≥0V ^⊗n the tensor algebra of V . It carries the multiplication induced by the canonical isomorphism V ^⊗r ⊗ V ^⊗s≅V ^⊗(r+s), making it into a unital associative algebra. The grading on T(V ) induced by the grading on V ^⊗n is given by $T {(V)}_{p} = \underset{i_{1} + … + i_{n} = p}{\oplus} V_{i_{1}} \otimes … \otimes V_{i_{n}}$ $Mathematical equation: \[ T(V)_p= \bigoplus_{i_1+\dotsc+i_n=p} V_{i_1}\otimes\dotsc\otimes V_{i_n} \]$ and is calledthe interior grading. On the other hand, T(V ) carries the grading given by T(V ) = ⊕ _n≥0V ^⊗n, which is called the exterior grading or grading by weight. If not specified otherwise, we always understand T(V ) to carry its interior grading. Note that both gradings make T(V ) into a graded algebra.

Example 8 (The symmetric and exterior algebra). Let I_S ⊂ T(V ) be the two-sided homogeneous ideal generated by elements of the form $v_{1} \otimes v_{2} - {(- 1)}^{| v_{1} | | v_{2} |} v_{2} \otimes v_{1}$ $Mathematical equation: $v_1\otimes v_2 - (-1)^{|v_1||v_2|}v_2\otimes v_1$$ . We call $S (V) ≔ T (V) / I_{S}$ $Mathematical equation: $\Sy(V)\vcentcolon= T(V)/I_S$$ the symmetric algebra of V . Similarly, the exterior algebra of V , denoted by ∧ V , is defined as the quotient of T(V ) by the two-sided homogeneous ideal generated by elements of the form $v_{1} \otimes v_{2} + {(- 1)}^{| v_{1} | | v_{2} |} v_{2} \otimes v_{1}$ $Mathematical equation: $v_1\otimes v_2 + (-1)^{|v_1||v_2|}v_2\otimes v_1$$ . We denote the multiplication in $S (V)$ $Mathematical equation: $\Sy(V)$$ and ∧ V by ∨ and ∧, respectively. Note that $S (V)$ $Mathematical equation: $\Sy(V)$$ and ∧ V also admit an exterior grading or grading by weight: as V ^⊗n ∩ I_S = U_S, we have $S (V) = \oplus_{n \geq 0} S^{n} (V)$ $Mathematical equation: $\Sy(V)=\bigoplus_{n\geq 0}\Sy^n(V)$$ and similarly ∧ V = ⊕ _n≥0 ∧ ⁿV.

It is easy to see that if A is a graded unital associative algebra, there is for each linear degree preserving map f : V → A a unique homomorphism of unital graded algebras φ: T(V ) → A that agrees on V with f (see for example [6], Proposition 1.1.1). It is then immediate that if A is commutative, φ factors to a unique homomorphism of unital graded algebras $S (V) \to A$ $Mathematical equation: $\Sy(V)\to A$$ . Applying this to the linear map $V \oplus W \to S (V) \otimes S (W), (v, w) \mapsto v \otimes 1 + 1 \otimes w$ $Mathematical equation: \[V\oplus W\to \Sy(V)\otimes \Sy(W), \quad (v,w)\mapsto v\otimes 1 + 1\otimes w\]$ yields a homomorphism of graded unital algebras $S (V \oplus W) \to S (V) \otimes S (W)$ $Mathematical equation: $\Sy(V\oplus W)\to \Sy(V)\otimes \Sy(W)$$ that is easilyseen to be an isomorphism with inverse $S (V) \otimes S (W) \to S (V \oplus W)$ $Mathematical equation: $\Sy(V)\otimes \Sy(W)\to \Sy(V\oplus W)$$ , v ⊗ w↦v ∨ w. With a slight modification of the sign in (6), similar arguments show that ∧ (V ⊕ W)≅∧ V ⊗∧ W. In particular, we have ${nn\Sy}^{n} (V \oplus W) ≅ \underset{p + q = n}{\oplus} S^{p} (V) \otimes S^{q} (W),$ $Mathematical equation: \begin{equation*} \Sy^n(V\oplus W)\cong \bigoplus_{p+q=n}\Sy^p(V)\otimes \Sy^q(W),\end{equation*}$ (7) ${nn\Asy}^{n} (V \oplus W) ≅ \underset{p + q = n}{\oplus} \Asy^{p} V \otimes \Asy^{q} W .$ $Mathematical equation: \begin{equation*} \Asy^n(V\oplus W)\cong \bigoplus_{p+q=n}\Asy^p V \otimes \Asy^q W. \end{equation*}$ (8)

For a graded algebra A, a derivation of A of degree p is a linear map d: A → A of degree p satisfying $d (a b) = d (a) b + {(- 1)}^{p | a |} a d (b)$ $Mathematical equation: \[ d(ab)=d(a)b+(-1)^{p|a|}ad(b) \]$

for all a, b ∈ A homogeneous. We denote by Der_p(A) the vector space of all derivations of A of degree p and by Der (A) the graded vector space $\underset{p \in ℤ}{\oplus} \Der_{p} (A)$ $Mathematical equation: $\bigoplus_{p\in\Z}\Der_p(A)$$ . A differential on the graded algebra A is an element d ∈Der(A) of degree one such that d² = 0. The pair (A, d) is then called a differential graded algebra (DG algebra for short). A homomorphism of DG algebras is a homomorphism of graded algebras that is also a homomorphism of DG vector spaces.

2.3 Graded Lie algebras and unshuffle permutations

Definition 9. A graded Lie algebra is a graded vector space L together with a (graded) antisymmetric degree preserving linear map [⋅, ⋅]: L ⊗ L → L called the Lie bracket satisfying the (graded) Jacobi identity $[x, [y, z]] = [[x, y], z] + {(- 1)}^{| x | | y |} [y, [x, z]] = 0$ $Mathematical equation: \begin{equation*}[x,[y,z]]=[[x,y],z]+(-1)^{|x||y|}[y,[x,z]]=0 \end{equation*}$ (9)

for all x, y, z ∈ L homogeneous.

If L is ungraded, we recover the usual definition of a Lie algebra. Note that (9) is nothing else than [x, ⋅ ] being a derivation of the graded algebra (L, [⋅, ⋅]).

Example 10. For a graded associative algebra A, we define the graded commutator [⋅, ⋅]: A ⊗ A → A by $[a, b] = a b - {(- 1)}^{| a | | b |} b a$ $Mathematical equation: $[a,b]= ab-(-1)^{|a||b|}ba$$ for a, b ∈ A homogeneous. This makes A into a graded Lie algebra. In particular, 𝔤𝔩(V ) := Hom(V, V ) becomes a graded Lie algebra. If V is itself a graded algebra, one can check that Der(V ) ⊂ 𝔤𝔩(V ) is a Lie subalgebra.

Definition 11. A differential graded Lie algebra (DGLA for short) is a DG algebra in which the underlying algebra is a graded Lie algebra.

Example 12. Let L be a graded Lie algebra and x ∈ L a degree one element such that $\frac{1}{2} [x, x] = 0$ $Mathematical equation: $\frac{1}{2}[x,x]=0$$ . Then d := [x, ⋅ ] satisfies d² = 0 by the Jacobi identity (9) and (L, [⋅, ⋅], d) is a DGLA. In particular, for (V, ∂) a DG vector space, this makes (𝔤𝔩(V ), [⋅, ⋅], [∂, ⋅ ]) canonically into a DGLA as $\frac{1}{2} [\partial, \partial] = \partial^{2} = 0$ $Mathematical equation: $\frac{1}{2}[\partial,\partial]=\partial^2=0$$ .

Definition 13. For a DGLA (L, [⋅, ⋅], d), a Maurer–Cartan element is an element x ∈ L of degree one such that $d (x) + \frac{1}{2} [x, x] = 0.$ $Mathematical equation: \begin{equation*}d(x)+\frac{1}{2}[x,x]=0. \end{equation*}$ (10)

The equation (10) is called the Maurer–Cartan equation.

Example 14. Let (L, [⋅, ⋅], d = [x, ⋅ ]) be as in Example 12. For y ∈ L of degree one, we then have $\frac{1}{2} [x + y, x + y] = 0$ $Mathematical equation: $\frac{1}{2}[x+y,x+y]=0$$ if and only if y satisfies the Maurer–Cartan equation.

For 0 ≤ i ≤ n, an (i, n− i)-unshuffle is a permutation σ ∈ 𝔖_n satisfying σ(1) < … < σ(i) and σ(i + 1) < … < σ(n). Following the notation in [3], we denote the set of all (i, n − i)-unshuffles by $\Sh_{i, n - i}^{- 1} \subset S_{n}$ $Mathematical equation: ${\Sh^{-1}_{{i,n-i}}}\subset\mathfrak S_n$$ . Using the antisymmetry of the Lie bracket, one can rewrite (9) as $\sum_{σ \in \Sh_{2, 1}^{- 1}} χ (σ) [[x_{σ (1)}, x_{σ (2)}], x_{σ (3)}] = 0$ $Mathematical equation: \begin{equation*}\sum_{\sigma\in\Sh^{-1}_{2,1}}\chi(\sigma)[[x_{\sigma(1)},x_{\sigma(2)}],x_{\sigma(3)}]=0 \end{equation*}$ (11)

for all x₁, x₂, x₃ ∈ L homogeneous.

Lemma 15. Each element σ ∈ 𝔖_n has for each i ∈{0, …, n} a unique decomposition σ = τ(α, β), where $τ \in \Sh_{i, n - i}^{- 1}$ $Mathematical equation: $\tau\in \Sh^{-1}_{i,n-i}$$ and (α, β) ∈ 𝔖_i × 𝔖_n−i. Here, 𝔖_i × 𝔖_n−i is considered as a subgroup of 𝔖_n in the obvious way.

Proof. Clearly, τ has to be the unique (i, n − i)-unshuffle such that {τ(1), …, τ(i)} = {σ(1), …, σ(i)} and {τ(i + 1), …, τ(n)} = {σ(i + 1), …, σ(n)}. We then have τ⁻¹σ ∈ 𝔖_i × 𝔖_n−i.

2.4 Graded coalgebras

A (graded) coalgebra (C, Δ) is a graded vector space C together with a degree preserving linear map Δ: C → C ⊗ C called the coproduct. If the diagram

commutes, C is called coassociative. We call C counital if there is a degree preserving linear map $ε : C \to k$ $Mathematical equation: $\varepsilon\colon C\to {\mathbb k}$$ such that the diagram

commutes. The map ε is then called the counit of C. If τ_C,C○Δ = Δ, then C is called cocommutative. A linear degree preserving map f : C → D between coalgebras (C, Δ_C) and (D, Δ_D) is called a homomorphism of coalgebras if $(f \otimes f) Δ_{C} = Δ_{D} ○ f .$ $Mathematical equation: \begin{equation*}(f\otimes f)\Delta_C=\Delta_D\circ f. \end{equation*}$ (14)

If C and D are counital with counits ε and η, respectively, and if f also satisfies η○f = ε, we call f a homomorphism of counital coalgebras.

For a coassociative coalgebra (C, Δ) and $n \in ℕ$ $Mathematical equation: $n\in \N$$ , we define the iterated coproduct Δⁿ: C → C^⊗(n+1) by Δ⁰ = id_C and Δⁿ = (Δ ⊗id_C ⊗… ⊗id_C)Δⁿ⁻¹ for n ≥ 1. It is convenient to then use Sweedler notation and to write Δⁿ(x) ∈ C^⊗(n+1) for x ∈ C as $Δ^{n} (x) = \sum x_{(1)} \otimes … \otimes x_{(n + 1)} .$ $Mathematical equation: \[\Delta^n(x)=\sum x_{(1)}\otimes\dotsc\otimes x_{(n+1)}.\]$

In this notation, for example, the condition for C to be cocommutative becomes $\sum x_{(1)} \otimes x_{(2)} = \sum {(- 1)}^{| x_{(1)} | | x_{(2)} |} x_{(2)} \otimes x_{(1)}$ $Mathematical equation: $\sum x_{(1)}\otimes x_{(2)}=\sum (-1)^{|x_{(1)}||x_{(2)}|} x_{(2)}\otimes x_{(1)}$$ for all x ∈ C.

Lemma 16. Let (C, Δ_C) be a coassociative coalgebra. Then for all $p, q \in ℕ$ $Mathematical equation: $p,q\in \N$$ , $(Δ_{C}^{p} \otimes Δ_{C}^{q}) Δ_{C} = Δ_{C}^{p + q + 1} .$ $Mathematical equation: \begin{equation*}(\Delta_C^p\otimes \Delta_C^q)\Delta_C=\Delta_C^{p+q+1}. \end{equation*}$ (15)

If (D, Δ_D) is another coassociative coalgebra and if f : C → D is a coalgebra homomorphism, $f^{\otimes (n + 1)} ○ Δ_{C}^{n} = Δ_{D}^{n} ○ f$ $Mathematical equation: \begin{equation*}f^{\otimes (n+1)}\circ \Delta_C^n=\Delta_D^n\circ f \end{equation*}$ (16) holds for all $n \in ℕ$ $Mathematical equation: $n\in\N$$ .

Proof. One obtains (15) and (16) by iterating (12) and (14); the details are left to the reader or can be found in ([7, Lemma-Definition VIII.10).

2.4.1 Coaugmented coalgebras

A coaugmented coalgebra (C, Δ, ε, u) is a counital coassociative coalgebra (C, Δ, ε) together with a homomorphism of counital coalgebras $u : k \to C$ $Mathematical equation: $u\colon{\mathbb k}\to C$$ . The coproduct on $k$ $Mathematical equation: ${\mathbb k}$$ is given by $1_{k} \mapsto 1_{k} \otimes 1_{k}$ $Mathematical equation: $1_{\mathbb k}\mapsto 1_{\mathbb k}\otimes 1_{\mathbb k}$$ and its counit is the identity on $k$ $Mathematical equation: ${\mathbb k}$$ . Denoting $u (1_{k}) \in C$ $Mathematical equation: $u(1_{\mathbb k})\in C$$ as 1, the conditions for u to be a homomorphism of counital coalgebras become Δ(1) = 1 ⊗ 1 and $ε ○ u = {id}_{k}$ $Mathematical equation: $\varepsilon\circ u=\id_{\mathbb k}$$ . A homomorphism between coaugmented coalgebras C and D is a homomorphism of counital coalgebras f : C → D such that f(1) = 1.

Given a coaugmented coalgebra (C, Δ, ε, u), set $\ol C ≔ \ker (ε)$ $Mathematical equation: $\ol C\vcentcolon =\ker(\varepsilon)$$ . We claim that $C ≅ \ol C \oplus k$ $Mathematical equation: $C\cong \ol C \oplus {\mathbb k}$$ . Indeed, as $ε ○ u = {id}_{k}$ $Mathematical equation: $\varepsilon\circ u=\id_{\mathbb k}$$ , the short exact sequence of graded vector spaces $0 \to \ker (ε) � C \overset{ε}{\to} k \to 0$ $Mathematical equation: \[0\to \ker(\varepsilon)\hookrightarrow C\xrightarrow{\varepsilon} {\mathbb k} \to 0\]$

splits. For $x \in \ol C$ $Mathematical equation: $x\in \ol C$$ , we then define $\ol Δ (x) ≔ Δ (x) - x \otimes 1 - 1 \otimes x$ $Mathematical equation: $\ol \Delta(x)\vcentcolon= \Delta(x)- x\otimes 1 -1 \otimes x$$ . Using (13) and ε(x) = 0, one easily sees that $\ol Δ (x) \in \ker (ε \otimes \id_{C}) \cap \ker ({id}_{C} \otimes ε) = \ol C \otimes \ol C;$ $Mathematical equation: \[ \ol\Delta(x)\in \ker(\varepsilon\otimes \id_C)\cap \ker(\id_C\otimes \varepsilon)= \ol C\otimes \ol C; \]$

for the last equality, note that $C \otimes C ≅ (\ol C \otimes \ol C) \oplus (\ol C \otimes k) \oplus (k \otimes \ol C) \oplus (k \otimes k) .$ $Mathematical equation: $C\otimes C\cong (\ol C\otimes \ol C) \oplus (\ol C\otimes {\mathbb k})\oplus ({\mathbb k}\otimes \ol C)\oplus ({\mathbb k}\otimes {\mathbb k}).$$ We call $\ol Δ : \ol C \to \ol C \otimes \ol C$ $Mathematical equation: $\ol \Delta\colon \ol C\to \ol C\otimes \ol C$$ the reduced coproduct on $\ol C$ $Mathematical equation: $\ol C$$ . It is straightforward to check that $(\ol C, \ol Δ)$ $Mathematical equation: $(\ol C,\ol\Delta)$$ is a coassociative coalgebra.

Conversely, given a coassociative coalgebra $(\ol C, \ol Δ)$ $Mathematical equation: $(\ol C, \ol \Delta)$$ , we define a coproduct on $C ≔ \ol C \oplus k$ $Mathematical equation: $C\vcentcolon= \ol C\oplus {\mathbb k}$$ by Δ(1) = 1 ⊗ 1 and $Δ (x) = \ol Δ (x) + x \otimes 1 + 1 \otimes x$ $Mathematical equation: $\Delta(x)=\ol \Delta(x) + x\otimes 1+1\otimes x$$ for $x \in \ol C$ $Mathematical equation: $x\in \ol C$$ . This makes C into a coaugmented coalgebra; the counit and coaugmentation map are given by the projection $C \to k$ $Mathematical equation: $C\to {\mathbb k}$$ and the inclusion $k � C$ $Mathematical equation: ${\mathbb k} \hookrightarrow C$$ , respectively.These constructions are clearly inverse to each other (up to isomorphism).

Let C and D be coaugmented coalgebras with counitsε and η, respectively. A linear degree preserving map f : C → D satisfying η○f = ε and f(1) = 1 decomposes as $f = \ol f \oplus \id_{k} : \ol C \oplus k \to \ol D \oplus k$ $Mathematical equation: \[f=\ol f\oplus \id_{\mathbb k}\colon \ol C \oplus {\mathbb k}\to \ol D \oplus {\mathbb k}\]$

for a unique degree preserving linear map $\ol f : \ol C \to \ol D$ $Mathematical equation: $\ol f\colon \ol C\to \ol D$$ . It is then easy to see that f is a homomorphism of coaugmented coalgebras if and only if $\ol f$ $Mathematical equation: $\ol f$$ is a homomorphism of coalgebras. This yields a one-to-one correspondence between coalgebra homomorphisms $\ol C \to \ol D$ $Mathematical equation: $\ol C\to \ol D$$ and homomorphisms of coaugmented coalgebras C → D.

Loosely speaking, this let us choose if we want to work with coaugmented coalgebras or non-coaugmented ones. In more technical terms, we have an equivalence between the category of coaugmented coalgebras and the category of coassociative coalgebras.

We call a coaugmented coalgebra (C, Δ) conilpotent if for all $x \in \ol C$ $Mathematical equation: $x\in \ol C$$ there is an $n \in ℕ$ $Mathematical equation: $n\in \N$$ such that $\ol Δ^{n} (x) = 0$ $Mathematical equation: $\ol\Delta^n (x)=0$$ .

2.4.2 Examples of coalgebras

There is a coproduct $\ol Δ_{A}$ $Mathematical equation: $\ol\Delta_A$$ on $\rten (V) ≔ \underset{n \geq 1}{\oplus} V^{\otimes n}$ $Mathematical equation: $\rten(V)\vcentcolon= \bigoplus_{n\geq 1}{V^{\otimes n}}$$ given by $\ol Δ_{A} (v_{1} \dots v_{n}) = \sum_{i = 1}^{n - 1} (v_{1} \dots v_{i}) \otimes (v_{i + 1} … v_{n}),$ $Mathematical equation: \[ \ol\Delta_A(v_1\dots v_n) = \sum_{i=1}^{n-1}(v_1\dots v_i)\otimes (v_{i+1}\dotsc v_n), \]$

where we now denote the multiplication in T(V ) by concatenation to avoid ambiguities. This makes $\rten (V)$ $Mathematical equation: $\rten(V)$$ into a coassociative graded coalgebra. The induced coaugmented coalgebra (T(V ), Δ_A) is called the tensor coalgebra. Inductively, one finds $\begin{array}{l} \ol Δ_{A}^{m} (v_{1} \dots v_{n}) \\ = \sum_{1 \leq i_{1} < … < i_{m} < n} (v_{1} \dots v_{i_{1}}) \otimes … \otimes (v_{i_{m} + 1} … v_{n}), \end{array}$ $Mathematical equation: \begin{eqnarray*}&&\hskip-6pt\ol\Delta_A^m(v_1\dots v_n)\nonumber\\ &&\quad=\sum_{1\leq i_1 <\dotsc<i_m<n}(v_1\dots v_{i_1})\otimes\dotsc\otimes (v_{i_m+1}\dotsc v_n),\nonumber\\ \end{eqnarray*}$ (17) which shows that T(V ) is conilpotent.

Proposition 17. Let (C, Δ) be a conilpotent coalgebra and $f : \ol C \to V$ $Mathematical equation: $f\colon \ol C\to V$$ a linear degree preserving map. There is a unique homomorphism of coaugmented coalgebras $\tilde{f} : C \to T (V)$ $Mathematical equation: $\tilde f\colon C \to T(V)$$ such that $f = \pr_{V} ○ \tilde{f},$ $Mathematical equation: $ f = \pr_V \circ\tilde f, $$ where here and subsequently, pr_{( ⋅)} denotes the projection onto a subspace under a given decomposition.

Proof. It clearly suffices to show that there is a unique homomorphism of coalgebras $\tilde{f} : \ol C \to \rten (V)$ $Mathematical equation: $\tilde f\colon \ol C\to \rten(V)$$ satifying $\pr_{V} ○ \tilde{f} = f$ $Mathematical equation: $\pr_V\circ \tilde f= f$$ . For the uniqueness, assume that there is such $\tilde{f}$ $Mathematical equation: $\tilde f$$ . By Lemma 16, we have ${\tilde{f}}^{\otimes (n + 1)} ○ \ol Δ^{n} = \ol Δ_{A}^{n} ○ \tilde{f}$ $Mathematical equation: \[ \tilde f^{\otimes(n+1)}\circ \ol\Delta^n=\ol\Delta_A^n \circ\tilde f \]$

for all $n \in ℕ$ $Mathematical equation: $n\in\N$$ . Composing both sides with $\pr_{V}^{\otimes (n + 1)}$ $Mathematical equation: $\pr_V^{\otimes(n+1)}$$ and noting that $\pr_{V}^{\otimes (n + 1)} ○ \ol Δ_{A}^{n} = \pr_{V^{\otimes (n + 1)}}$ $Mathematical equation: $\pr_V^{\otimes(n+1)}\circ\ol\Delta_A^n= \pr_{V^{\otimes(n+1)}}$$ then yields $\pr_{V^{\otimes (n + 1)}} ○ \tilde{f} = \pr_{V}^{\otimes (n + 1)} ○ {\tilde{f}}^{\otimes (n + 1)} ○ \ol Δ^{n} = f^{\otimes (n + 1)} ○ \ol Δ^{n} .$ $Mathematical equation: \[ \pr_{V^{\otimes(n+1)}} \circ\tilde f=\pr_V^{\otimes(n+1)}\circ\tilde f^{\otimes(n+1)}\circ\ol\Delta^n= f^{\otimes(n+1)} \circ\ol\Delta^n. \]$

This shows that $\tilde{f}$ $Mathematical equation: $\tilde f$$ is completely determined by f and therefore unique. For the existence, consider the linear map $\sum_{n = 0}^{\infty} \ol Δ^{n} : \ol C \to \rten (\ol C) .$ $Mathematical equation: $\sum_{n=0}^{\infty} \ol\Delta^n \colon \ol C \to \rten(\ol C).$$ This is well-defined as C is conilpotent. A straightforward computation using Lemma 16 and ${\ol Δ_{A} |}_{V} = 0$ $Mathematical equation: $\left.\ol\Delta_A\right|_{V}=0$$ shows that $(\sum_{n = 0}^{\infty} \ol Δ^{n} \otimes \sum_{n = 0}^{\infty} \ol Δ^{n}) \ol Δ = \ol Δ_{A} ○ \sum_{n = 0}^{\infty} \ol Δ^{n},$ $Mathematical equation: \[ \left(\sum_{n=0}^{\infty} \ol\Delta^n \otimes \sum_{n=0}^{\infty} \ol\Delta^n \right)\ol\Delta =\ol\Delta_A \circ\sum_{n=0}^{\infty} \ol\Delta^{n}, \]$

so that $\sum_{n = 0}^{\infty} \ol Δ^{n}$ $Mathematical equation: $\sum_{n=0}^{\infty} \ol\Delta^n$$ is a coalgebra homomorphism. As $\rten (f) = \underset{n \geq 1}{\oplus} f^{\otimes n} : \rten (\ol C) \to \rten (V)$ $Mathematical equation: $\rten(f)=\bigoplus_{n\geq 1}f^{\otimes n}\colon \rten(\ol C)\to \rten(V)$$ is easily seen to be also a coalgebra homomorphism, $\tilde{f} ≔ \rten (f) ○ \sum_{n = 0}^{\infty} \ol Δ^{n} : C \to \rten (V)$ $Mathematical equation: $\tilde f \vcentcolon = \rten(f) \circ\sum_{n=0}^{\infty} \ol\Delta^n \colon C \to \rten(V)$$ is a homomorphism of coalgebras with $\pr_{V} ○ \tilde{f} = f$ $Mathematical equation: $\pr_V\circ \tilde f= f$$ .

Let $\rsym (V) ≔ \underset{n \geq 1}{\oplus} S^{n} (V)$ $Mathematical equation: $\rsym(V)\vcentcolon= \bigoplus_{n\geq 1}\Sy^n(V)$$ . Consider the linear maps $\begin{array}{l} π : \rten (V) \to \rsym (V), v_{1} \otimes … \otimes v_{n} \mapsto \frac{1}{n!} v_{1} \lor … \lor v_{n}, \\ N : \rsym (V) \to \rten (V), \\ v_{1} \lor … \lor v_{n} \mapsto \sum_{σ \in S_{n}} ε (σ) v_{σ (1)} \otimes … \otimes v_{σ (n)} . \end{array}$ $Mathematical equation: \[ \begin{array}{l} \displaystyle\pi\colon \rten(V)\to \rsym(V), \quad v_1\otimes\dotsc\otimes v_n\mapsto \frac{1}{n!} v_1\vee\dotsc\vee v_n,\\[5pt] \displaystyle N\colon \rsym(V)\to \rten(V),\nonumber\\[5pt] \displaystyle v_1 \vee \dotsc\vee v_n\mapsto \sum_{\sigma\in \mathfrak S_n}\varepsilon(\sigma)v_{\sigma(1)}\otimes\dotsc\otimes v_{\sigma(n)}. \end{array} \]$

It is immediate that π○N = id_S, where here and subsequently, we abbreviate $\rsym (V)$ $Mathematical equation: $\rsym(V)$$ and $S (V)$ $Mathematical equation: $\Sy(V)$$ to S in subscripts. Using Lemma 15, we compute $\begin{array}{l} \ol Δ_{A} (N (v_{1} \lor \dots \lor v_{n})) \\ = \sum_{i = 1}^{n - 1} \sum_{σ \in S_{n}} ε (σ) (v_{σ (1)} … v_{σ (i)}) \otimes (v_{σ (i + 1)} … v_{σ (n)}) \\ = \sum_{i = 1}^{n - 1} \sum_{τ \in \Sh_{i, n - i}^{- 1}} \sum_{(α, β) \in S_{i} \times S_{n - i}} ε (τ (α, β)) (v_{(τ (α, β)) (1)} … v_{(τ (α, β)) (i)}) \otimes (v_{(τ (α, β)) (i + 1)} … v_{(τ (α, β)) (n)}) \\ = \sum_{i = 1}^{n - 1} \sum_{τ \in \Sh_{i, n - i}^{- 1}} ε (τ) \sum_{(α, β) \in S_{i} \times S_{n - i}} \hat{ε} (α) (v_{τ (1)} … v_{τ (i)}) \otimes \hat{ε} (β) (v_{τ (i + 1)} … v_{τ) (n)}) \\ = \sum_{i = 1}^{n - 1} \sum_{τ \in \Sh_{i, n - i}^{- 1}} ε (τ) N (v_{1} \lor … \lor v_{i}) \otimes N (v_{i + 1} \lor … \lor v_{n}) . \end{array}$ $Mathematical equation: \[ \begin{array}{ll}\displaystyle &\displaystyle\ol\Delta_A(N(v_1\vee\dots\vee v_n)) \\ &\displaystyle=\sum_{i=1}^{n-1} \sum_{\sigma\in \mathfrak S_n}\varepsilon(\sigma)(v_{\sigma(1)}\dotsc v_{\sigma(i)})\otimes(v_{\sigma(i+1)}\dotsc v_{\sigma(n)})\\ &\displaystyle=\sum_{i=1}^{n-1}\sum_{\tau\in\Sh^{-1}_{i,n-i}}\sum_{(\alpha,\beta)\in\mathfrak S_i \times \mathfrak S_{n-i}}\varepsilon(\tau(\alpha,\beta)) (v_{(\tau(\alpha,\beta))(1)}\dotsc v_{(\tau(\alpha,\beta))(i)})\otimes (v_{(\tau(\alpha,\beta))(i+1)}\dotsc v_{(\tau(\alpha,\beta))(n)})\\ &\displaystyle=\sum_{i=1}^{n-1}\sum_{\tau\in\Sh^{-1}_{i,n-i}}\varepsilon(\tau)\sum_{(\alpha,\beta)\in\mathfrak S_i \times \mathfrak S_{n-i}}\hat\varepsilon(\alpha)(v_{\tau(1)}\dotsc v_{\tau(i)})\otimes \hat\varepsilon(\beta)(v_{\tau(i+1)}\dotsc v_{\tau)(n)})\\ &\displaystyle= \sum_{i=1}^{n-1}\sum_{\tau\in\Sh^{-1}_{i,n-i}}\varepsilon(\tau)N(v_1\vee\dotsc\vee v_i)\otimes N(v_{i+1}\vee\dotsc\vee v_n). \end{array} \]$

This shows that $\ima (N) \subset \rten (V)$ $Mathematical equation: $\ima(N)\subset \rten(V)$$ is a subcoalgebra and induces a coproduct on $\rsym (V) ≅ \ima (N)$ $Mathematical equation: $\rsym(V)\cong \ima(N)$$ . As a subcoalgebra of a coassociative coalgebra is clearly coassociative itself, $S (V)$ $Mathematical equation: $\Sy(V)$$ becomes a coaugmented coalgebra with the coproduct $Δ_{S} : S (V) \to S (V) \otimes S (V)$ $Mathematical equation: $\Delta_S\colon \Sy(V)\to \Sy(V)\otimes \Sy(V)$$ given by $\begin{array}{l} Δ_{S} (v_{1} \lor … \lor v_{n}) & = & \sum_{i = 0}^{n} \sum_{τ \in \Sh_{i, n - i}^{- 1}} ε (τ) (v_{τ (1)} \lor … \lor v_{τ (i)}) \\ \otimes (v_{τ (i + 1)} \lor … \lor v_{τ (n)}) . \end{array}$ $Mathematical equation: \begin{eqnarray*}\Delta_S(v_1\vee\dotsc\vee v_n)&=&\sum_{i=0}^{n}\sum_{\tau\in\Sh^{-1}_{i,n-i}}\varepsilon(\tau)(v_{\tau(1)}\vee\dotsc \vee v_{\tau(i)})\nonumber\\ &&\otimes\,(v_{\tau(i+1)}\vee\dotsc\vee v_{\tau(n)}). \end{eqnarray*}$ (18)

It is immediate that $S (V)$ $Mathematical equation: $\Sy(V)$$ is conilpotent, as it is a subcoalgebra of T(V ). We claim that $S (V)$ $Mathematical equation: $\Sy(V)$$ is even cocommutative. Indeed, let σ_i ∈ 𝔖_n be for 0 ≤ i ≤ n the permutation given by (σ_i(1), …, σ_i(n)) = (i + 1, …, n, 1, …, i). We then have $\Sh_{i, n - i}^{- 1} σ_{i} = \Sh_{n - i, i}^{- 1}$ $Mathematical equation: $\Sh^{-1}_{i,n-i}\sigma_i=\Sh^{-1}_{n-i,i}$$ and therefore $\begin{array}{l} τ_{S, S} ○ Δ_{S} (v_{1} \lor … \lor v_{n}) \\ = \sum_{i = 0}^{n} \sum_{τ \in \Sh_{i, n - i}^{- 1}} ε (τ σ_{i}) (v_{(τ σ_{i}) (1)} \lor … \lor v_{(τ σ_{i}) (n - i)}) \\ \otimes (v_{(τ σ_{i}) (n - i + 1)} \lor … \lor v_{(τ σ_{i}) (n)}) \\ = \sum_{i = 0}^{n} \sum_{τ \in \Sh_{n - i, i}^{- 1}} ε (τ) (v_{τ (1)} \lor … \lor v_{τ (i)}) \\ \otimes (v_{τ (i + 1)} \lor … \lor v_{τ (n)}) \\ = Δ_{S} (v_{1} \lor … \lor v_{n}) . \end{array}$ $Mathematical equation: \[ \begin{array}{ll} &\displaystyle\tau_{S,S}\circ\Delta_S(v_1\vee\dotsc\vee v_n)\\&\displaystyle=\sum_{i=0}^{n}\sum_{\tau\in\Sh^{-1}_{i,n-i}}\varepsilon(\tau\sigma_i)(v_{(\tau\sigma_i)(1)}\vee\dotsc \vee v_{(\tau\sigma_i)(n-i)})\\ &\displaystyle\quad\otimes(v_{(\tau\sigma_i)(n-i+1)}\vee\dotsc\vee v_{(\tau\sigma_i)(n)})\\ &\displaystyle=\sum_{i=0}^{n}\sum_{\tau\in\Sh^{-1}_{n-i,i}}\varepsilon(\tau)(v_{\tau(1)}\vee\dotsc \vee v_{\tau(i)})\\ &\displaystyle\quad \otimes(v_{\tau(i+1)}\vee\dotsc\vee v_{\tau(n)})\\ &\displaystyle=\Delta_S(v_1\vee\dotsc\vee v_n). \end{array} \]$

Proposition 18. Let (C, Δ) be a cocommutative conilpotent coalgebra and $f : \ol C \to V$ $Mathematical equation: $f\colon \ol C\to V$$ a degree preserving linear map. There is a unique homomorphism of coaugmented coalgebras $\tilde{f} : C \to S (V)$ $Mathematical equation: $\tilde f\colon C \to \Sy(V)$$ such that $f = \pr_{V} ○ \tilde{f} .$ $Mathematical equation: $ f = \pr_V \circ\tilde f. $$

Proof. As in the proof of Proposition 17, it suffices to show that there is a unique homomorphism of coalgebras $\tilde{f} : \ol C \to \rsym (V)$ $Mathematical equation: $\tilde f\colon \ol C\to \rsym(V)$$ satisfying $\pr_{V} ○ \tilde{f} = f$ $Mathematical equation: $\pr_V\circ \tilde f=f$$ . Recall that $\ol T (f) ○ \sum_{n} \ol Δ^{n}$ $Mathematical equation: $\ol T(f)\circ \sum_n \ol\Delta^n$$ is the unique coalgebra homomorphism $\ol C \to \ol T (V)$ $Mathematical equation: $\ol C\to \ol T(V)$$ extending f. For 0 ≤ i ≤ n − 1, we have $\begin{array}{l} ({id}_{V}^{\otimes i} \otimes τ_{C, C} \otimes \id_{V}^{\otimes (n - i - 1)}) \rten (f) ○ \ol Δ^{n} \\ = \rten (f) ({id}_{C}^{\otimes i} \otimes τ_{C, C} \otimes \id_{C}^{\otimes (n - i - 1)}) \ol Δ^{n} \\ = \rten (f) ({id}_{C}^{\otimes i} \otimes (τ_{C, C} ○ \ol Δ) \otimes \id_{C}^{\otimes (n - i - 1)}) \ol Δ^{n - 1} \\ = \rten (f) ({id}_{C}^{\otimes i} \otimes \ol Δ \otimes \id_{C}^{\otimes (n - i - 1)}) \ol Δ^{n - 1} \\ = \rten (f) ○ \ol Δ^{n} \end{array}$ $Mathematical equation: \[\begin{array}{ll} &\displaystyle(\id_V^{\otimes i}\otimes \tau_{C,C}\otimes \id_V^{\otimes(n-i-1)})\rten(f)\circ\ol\Delta^n\\[4pt] &\displaystyle\quad=\rten(f)(\id_C^{\otimes i}\otimes \tau_{C,C}\otimes \id_C^{\otimes(n-i-1)})\ol\Delta^n\\[4pt] &\displaystyle\quad= \rten(f)(\id_C^{\otimes i}\otimes (\tau_{C,C}\circ\ol\Delta) \otimes \id_C^{\otimes(n-i-1)})\ol\Delta^{n-1}\\[4pt] &\displaystyle\quad= \rten(f)(\id_C^{\otimes i}\otimes \ol\Delta \otimes \id_C^{\otimes(n-i-1)})\ol\Delta^{n-1}\\[4pt] &\displaystyle\quad= \rten(f)\circ\ol\Delta^{n} \end{array} \]$

since C is cocommutative. As $({id}_{V}^{\otimes i} \otimes τ_{C, C} \otimes \id_{V}^{\otimes (n - i - 1)}) = \hat{ε} (s_{i}),$ $Mathematical equation: $(\id_V^{\otimes i}\otimes \tau_{C,C}\otimes \id_V^{\otimes(n-i-1)})= \hat\varepsilon(s_{i}),$$ the image of $\rten (f) ○ \sum_{n} \ol Δ^{n}$ $Mathematical equation: $\rten(f)\circ \sum_n\ol\Delta^n$$ is contained in the subspace of $\rten (V)$ $Mathematical equation: $\rten(V)$$ of all symmetric elements, which is im(N). We obtain an induced homomorphism of coalgebras $\begin{array}{l} \tilde{f} = π ○ \ol T (f) ○ \sum_{n = 0}^{\infty} \ol Δ^{n} : \ol C \to \rsym (V), \\ x \mapsto \sum_{n = 1}^{\infty} \frac{1}{n!} \sum f (x_{(1)}) \lor … \lor f (x_{(n)}) \end{array}$ $Mathematical equation: \begin{equation*}\begin{array}{l} \displaystyle\tilde f=\pi\circ \ol T(f)\circ \sum_{n=0}^{\infty}\ol\Delta^n \colon \ol C\to \rsym(V),\\[4pt] \displaystyle x\mapsto \sum_{n=1}^{\infty} \frac{1}{n!}\sum f(x_{(1)})\vee\dotsc\vee f(x_{(n)}) \end{array} \end{equation*}$ (19)

with $\pr_{V} ○ \tilde{f} = f$ $Mathematical equation: $\pr_V \circ\tilde f= f$$ . Similarly, a coalgebra homomorphism $\tilde{f} : \ol C \to \rsym (V)$ $Mathematical equation: $\tilde f\colon \ol C \to \rsym(V)$$ gives rise to a coalgebra homomorphism $N ○ \tilde{f} : \ol C \to \rten (V)$ $Mathematical equation: $N \circ \tilde f \colon \ol C\to \rten(V)$$ that is uniquely determined by $\pr_{V} ○ N ○ \tilde{f} = \pr_{V} ○ \tilde{f}$ $Mathematical equation: $\pr_V \circ N\circ\tilde f=\pr_V\circ \tilde f$$ by Proposition 17. As N is injective, this shows uniqueness of $\tilde{f}$ $Mathematical equation: $\tilde f$$ .

Example 19. A linear degree preserving map f : V → W can be extended by zero to a linear map $\rsym (V) \to W$ $Mathematical equation: $\rsym(V)\to W$$ . The induced homomorphism of coaugmented coalgebras $S (V) \to S (W)$ $Mathematical equation: $\Sy(V)\to \Sy(W)$$ is denoted by $S (f)$ $Mathematical equation: $\Sy(f)$$ and is given by $S (f) (v_{1} \lor … \lor v_{n}) = f (v_{1}) \lor … \lor f (v_{n}) .$ $Mathematical equation: $ \Sy(f)(v_1\vee\dotsc\vee v_n)=f(v_1)\vee\dotsc\vee f(v_n). $$

2.4.3 Comodules and coderivations

Let (C, Δ) be a coassociative coalgebra. A left comodule over C is a graded vector space M together with a degree preserving linear map Δ_l: M → C ⊗ M satisfying $(Δ \otimes \id_{M}) Δ_{l} = ({id}_{C} \otimes Δ_{l}) Δ_{l} .$ $Mathematical equation: \begin{equation*}(\Delta \otimes \id_M)\Delta_l = (\id_C \otimes \Delta_l)\Delta_l. \end{equation*}$ (20)

Similarly, a right comodule over C is a graded vector space M together with a degree preserving linear map Δ_r: M → M ⊗ C such that $({id}_{M} \otimes Δ) Δ_{r} = (Δ_{r} \otimes \id_{C}) Δ_{r} .$ $Mathematical equation: \begin{equation*}(\id_M \otimes \Delta)\Delta_r = (\Delta_r\otimes \id_C)\Delta_r. \end{equation*}$ (21)

If M is both a left and a right comodule over C and if the compatibility relation $(Δ_{l} \otimes \id_{C}) Δ_{r} = ({id}_{C} \otimes Δ_{r}) Δ_{l}$ $Mathematical equation: \begin{equation*}(\Delta_l \otimes \id_C)\Delta_r = (\id_C \otimes \Delta_r)\Delta_l \end{equation*}$ (22)

is satisfied, M is called a (bi)comodule over C. Given such M, we define a coderivation of degree p to be a homogeneous linear map d: M → C of degree p such that $Δ ○ d = (d \otimes \id_{C}) Δ_{r} + ({id}_{C} \otimes d) Δ_{l} .$ $Mathematical equation: \begin{equation*}\Delta \circ d= (d\otimes \id_C)\Delta_r + (\id_C\otimes d)\Delta_l. \end{equation*}$ (23)

We denote the vector space of all these maps by Coder_p(M, C) and by Coder(M, C) the graded vector space $\underset{p \in ℤ}{\oplus} \Coder_{p} (M, C)$ $Mathematical equation: $\bigoplus_{p\in \Z} \Coder_p(M,C)$$ .

Let (C, Δ_C) and (D, Δ_D) be coassociative coalgebras and f : D → C a coalgebra homomorphism. Then Δ_r := (id_D ⊗ f)Δ_D and Δ_l := (f ⊗id_D)Δ_D make D into a comodule over C. In particular, C is a comodule over itself and we abbreviate Coder(C, C) to Coder(C). If C and D are coaugmented and if f is a homomorphism of coaugmented coalgebras, the comodule structure is compatible with the counit in the sense that the diagram

commutes, where ε is the counit on C. Observe that $\ol f : \ol D \to \ol C$ $Mathematical equation: $\ol f\colon \ol D \to \ol C$$ then makes $\ol D$ $Mathematical equation: $\ol D$$ into a comodule over $\ol C$ $Mathematical equation: $\ol C$$ . The following proposition relates elements in $\Coder (\ol D, \ol C)$ $Mathematical equation: $\Coder(\ol D,\ol C)$$ to coderivations d: D → C that satisfy d(1) = 0; the latter is called a coderivation of coaugmented coalgebras.

Proposition 20. Let f : D → C be a homomorphism of coaugmented coalgebras. There is a one-to-one correspondence between coderivations d: D → C satisfying d(1) = 0 and coderivations $\ol d : \ol D \to \ol C$ $Mathematical equation: $\ol d\colon \ol D\to \ol C$$ given by $d = \ol d \oplus 0 : \ol D \oplus k \to \ol C \oplus k$ $Mathematical equation: $d=\ol d\oplus 0\colon \ol D\oplus {\mathbb k} \to \ol C\oplus {\mathbb k}$$ .

Proof. Given a linear map $\ol d \in Hom (\ol D, \ol C)$ $Mathematical equation: $\ol d\in\Hom(\ol D,\ol C)$$ , one easily checksthat $\ol d$ $Mathematical equation: $\ol d$$ is a coderivation if and only if $\ol d \oplus 0 : \ol D \oplus k \to \ol C \oplus k$ $Mathematical equation: $\ol d\oplus 0\colon \ol D\oplus {\mathbb k} \to \ol C\oplus {\mathbb k}$$ is. It then suffices to show thateach coderivation d: D → C with d(1) = 0 is of this form. Let ε be the counit of C and $μ_{k} : k \otimes k \to k$ $Mathematical equation: $\mu_{\mathbb k}\colon {\mathbb k}\otimes {\mathbb k}\to {\mathbb k}$$ the multiplication on $k$ $Mathematical equation: ${\mathbb k}$$ . From (13), (23) and the compatibility with the counit (24) it then follows that $\begin{array}{l} ε ○ d & = μ_{k} (ε \otimes ε) Δ_{C} ○ d \\ = μ_{k} (ε \otimes (ε ○ d)) Δ_{l} + μ_{k} ((ε ○ d) \otimes ε) Δ_{r} \\ = 2 (ε ○ d), \end{array}$ $Mathematical equation: \begin{align*} \varepsilon\circ d&=\mu_{\mathbb k} (\varepsilon\otimes\varepsilon)\Delta_C\circ d\\ &=\mu_{\mathbb k} (\varepsilon\otimes (\varepsilon\circ d))\Delta_l+\mu_{\mathbb k} ((\varepsilon\circ d)\otimes\varepsilon)\Delta_r\\ &= 2(\varepsilon\circ d), \end{align*}$

which shows that $d (D) \subset \ol C$ $Mathematical equation: $d(D)\subset \ol C$$ . Hence, d decomposes as $\ol d \oplus 0 : \ol D \oplus k \to \ol C \oplus k$ $Mathematical equation: $\ol d\oplus 0\colon \ol D\oplus {\mathbb k}\to \ol C\oplus {\mathbb k}$$ .

For a coassociative coalgebra C, we call an element d ∈ Coder(C) of degree one with d² = 0 a codifferential on C. We then call the pair (C, d) a differential graded coassociative coalgebra (DGC for short). If C is coaugmented and d(1) = 0, we call (C, d) a coaugmented DGC. A homomorphism of DGCs is then a coalgebra homomorphism that is also a homomorphism of DG vector spaces; homomorphisms of coaugmented DGCs are defined accordingly. From Proposition 20 and Section 2.4.1, we then obtain an equivalence between the categories of DGCs and coaugmented DGCs.

Proposition 21. Let C be a coassociative coalgebra. Then Coder(C) ⊂ 𝔤𝔩(C) is closed under the graded commutator. Also, if f : D → C is a homomorphism of coassociative coalgebras, d ∈ Coder(C) and d′∈ Coder(D), then f○d′, d○f ∈ Coder(D, C).

Proof. Both parts of the proposition are straightforward computations which are left to the reader.

Theorem 22. Let D be a cocommutative coaugmented coalgebra and $f : D \to S (V)$ $Mathematical equation: $f\colon D\to \Sy(V)$$ a homomorphism of coaugmented coalgebras. The linear map $\Coder (D, S (V)) \to \Hom (D, V), d \mapsto \pr_{V} ○ d$ $Mathematical equation: \[\Coder(D,\Sy(V))\to \Hom(D,V), \quad d\mapsto \pr_V\circ d \]$ is then an isomorphism. Its inverse is given by $Hom (D, V) \to \Coder (D, S (V)), λ \mapsto μ_{S} (λ \otimes f) Δ_{D},$ $Mathematical equation: \[ \Hom(D,V) \to \Coder(D,\Sy(V)), \quad \lambda \mapsto \mu_S(\lambda\otimes f)\Delta_D, \]$ where $μ_{S} : S (V) \otimes S (V) \to S (V)$ $Mathematical equation: $\mu_S\colon \Sy(V)\otimes \Sy(V)\to \Sy(V)$$ denotes the multiplication on $S (V)$ $Mathematical equation: $\Sy(V)$$ and Δ_D the coproduct on D.

It is immediate that d(1) = 0 if and only if λ = pr_V○d vanishes on $k$ $Mathematical equation: ${\mathbb k}$$ . Together with Proposition 20, this shows $\Coder (\ol D, \rsym (V)) ≅ \Hom (\ol D, V)$ $Mathematical equation: $\Coder(\ol D,\rsym(V))\cong \Hom(\ol D,V)$$ .

The first part of Theorem 22 actually holds for a broader class of comodules over $S (V)$ $Mathematical equation: $\Sy(V)$$ (see for example [8], Lemma 2.4); the inverse formula d = μ_S(λ ⊗id_M)Δ_r then continues to hold for comodules M in which τ_M,S▼Δ_r = Δ_l.

Remark 23. For 1 ≤ i ≤ n − 1 and $τ \in \Sh_{i, n - i}^{- 1}$ $Mathematical equation: $\tau\in\Sh^{-1}_{i,n-i}$$ either τ(i) = n or τ(n) = n. In the first case, there is a unique $σ \in \Sh_{i - 1, n - i}^{- 1}$ $Mathematical equation: $\sigma\in\Sh^{-1}_{i-1,n-i}$$ such that (τ(1), …, τ(n)) = (σ(1), …, σ(i − 1), n, σ(i), …, σ(n − 1)), while in the second case (τ(1), …, τ(n)) = (σ(1), …, σ(n − 1), n) for a unique $σ \in \Sh_{i, n - i - 1}^{- 1}$ $Mathematical equation: $\sigma\in\Sh^{-1}_{i,n-i-1}$$ . This yields abijection $\Sh_{i, n - i}^{- 1} ≅ \Sh_{i - 1, n - i}^{- 1} ⊔ \Sh_{i, n - i - 1}^{- 1}$ $Mathematical equation: $\Sh^{-1}_{i,n-i}\cong\Sh^{-1}_{i-1,n-i}\sqcup \Sh^{-1}_{i,n-i-1}$$ . By setting $\Sh_{- 1, n}^{- 1}, \Sh_{n, - 1}^{- 1} = \emptyset$ $Mathematical equation: $\Sh^{-1}_{-1,n},\Sh^{-1}_{n,-1}=\emptyset$$ , this also holds for i = 0, n.

Lemma 24. The map $Δ_{S} : S (V) \to S (V) \otimes S (V)$ $Mathematical equation: $\Delta_S\colon \Sy(V)\to \Sy(V)\otimes \Sy(V)$$ is a homomorphism of graded algebras.

Proof. We show by induction over $n \in ℕ$ $Mathematical equation: $n\in\N$$ that $Δ_{S} (v_{1} \lor … \lor v_{n}) = Δ_{S} (v_{1}) … Δ_{S} (v_{n}) .$ $Mathematical equation: \begin{equation*}\Delta_S(v_1 \vee\dotsc \vee v_n)=\Delta_S(v_1)\dotsc \Delta_S(v_n). \end{equation*}$ (25)

For n = 1 there is nothing to do. Assume that (25) holds for n ≥ 1. We compute $\begin{array}{l} Δ_{S} (v_{1}) … Δ_{S} (v_{n + 1}) \\ = Δ_{S} (v_{1} \lor … \lor v_{n}) Δ_{S} (v_{n + 1}) \\ = \sum_{i = 0}^{n} \sum_{τ \in \Sh_{i, n - i}^{- 1}} (ε (τ) (v_{τ (1)} \lor … \lor v_{τ (i)}) \otimes (v_{τ (i + 1)} \lor … \lor v_{τ (n)})) (v_{n + 1} \otimes 1 + 1 \otimes v_{n + 1}) \\ = \sum_{i = 0}^{n} \sum_{σ \in \Sh_{i, n - i}^{- 1}} {(- 1)}^{| v_{n + 1} | \sum_{k = i + 1}^{n} | v_{σ (k)} |} ε (σ) (v_{σ (1)} \lor … \lor v_{σ (i)} \lor v_{n + 1}) \otimes (v_{σ (i + 1)} \lor … \lor v_{σ (n)}) \\ + \sum_{i = 0}^{n} \sum_{σ \in \Sh_{i, n - i}^{- 1}} ε (σ) (v_{σ (1)} \lor … \lor v_{σ (i)}) \otimes (v_{σ (i + 1)} \lor … \lor v_{σ (n)} \lor v_{n + 1}) \\ = \sum_{i = 0}^{n + 1} \sum_{σ \in \Sh_{i, n + 1 - i}^{- 1}} ε (σ) (v_{σ (1)} \lor … \lor v_{σ (i)}) \otimes (v_{σ (i + 1)} \lor … \lor v_{σ (n + 1)}) \\ = Δ_{S} (v_{1} \lor … \lor v_{n + 1}) . \end{array}$ $Mathematical equation: \[\begin{array}{ll} &\displaystyle\Delta_S(v_1)\dotsc \Delta_S(v_{n+1})\\ &\displaystyle=\Delta_S(v_1\vee\dotsc\vee v_n)\Delta_S(v_{n+1})\\ &\displaystyle=\sum_{i=0}^{n}\sum_{\tau\in\Sh^{-1}_{i,n-i}}\left(\varepsilon(\tau)(v_{\tau(1)}\vee\dotsc \vee v_{\tau(i)})\otimes(v_{\tau(i+1)}\vee\dotsc\vee v_{\tau(n)})\right)(v_{n+1}\otimes 1+1\otimes v_{n+1})\\ &\displaystyle=\sum_{i=0}^{n}\sum_{\sigma\in\Sh^{-1}_{i,n-i}}(-1)^{|v_{n+1}|\sum_{k=i+1}^n|v_{\sigma(k)}|}\varepsilon(\sigma) (v_{\sigma(1)}\vee\dotsc\vee v_{\sigma(i)}\vee v_{n+1})\otimes (v_{\sigma(i+1)}\vee\dotsc\vee v_{\sigma(n)})\\ &\displaystyle\quad+\sum_{i=0}^n\sum_{\sigma\in\Sh^{-1}_{i,n-i}}\varepsilon(\sigma)(v_{\sigma(1)}\vee\dotsc\vee v_{\sigma(i)})\otimes (v_{\sigma(i+1)}\vee\dotsc\vee v_{\sigma(n)}\vee v_{n+1})\\ &\displaystyle=\sum_{i=0}^{n+1}\sum_{\sigma\in\Sh^{-1}_{i,n+1-i}}\varepsilon(\sigma)(v_{\sigma(1)}\vee\dotsc\vee v_{\sigma(i)})\otimes (v_{\sigma(i+1)}\vee\dotsc\vee v_{\sigma(n+1)})\\ &\displaystyle=\Delta_S(v_1\vee\dotsc \vee v_{n+1}). \end{array}\]$

In the fourth equality, we shifted the summation index of the first sum and used Remark 23.

Proof of Theorem 22. Let $d : D \to S (V)$ $Mathematical equation: $d\colon D\to \Sy(V)$$ be a coderivation, that is $Δ_{S} ○ d = (d \otimes f + f \otimes d) Δ_{D} .$ $Mathematical equation: \[ \Delta_S\circ d = (d\otimes f + f\otimes d)\Delta_D. \]$

Inductively,we then get $Δ_{S}^{n} ○ d = \sum_{k = 0}^{n} (f^{\otimes k} \otimes d \otimes f^{\otimes (n - k)}) Δ_{D}^{n} .$ $Mathematical equation: \[ \Delta_S^n\circ d=\sum_{k=0}^n (f^{\otimes k}\otimes d\otimes f^{\otimes(n-k)})\Delta_D^n. \]$

For $n \in ℕ$ $Mathematical equation: $n\in\N$$ , let $π_{n} : T^{n} (V) \to S^{n} (V)$ $Mathematical equation: $\pi_n\colon T^n(V)\to \Sy^n(V)$$ be the linear map defined by $π_{n} (v_{1} \otimes … \otimes v_{n}) = \frac{1}{n!} v_{1} \lor … \lor v_{n} .$ $Mathematical equation: \[\pi_n(v_1\otimes\dotsc\otimes v_n)=\frac{1}{n!}v_1\vee\dotsc\vee v_n.\]$

From $S (V)$ $Mathematical equation: $\Sy(V)$$ being a subcoalgebra of T(V ) and (17), it follows that $π_{n + 1} ○ \pr_{V}^{\otimes (n + 1)} ○ Δ_{S}^{n} = \pr_{S^{n + 1} (V)}$ $Mathematical equation: $\pi_{n+1}\circ \pr_V^{\otimes (n+1)}\circ \Delta_S^{n}=\pr_{\Sy^{n+1}(V)}$$ . We then have $\begin{array}{l} \pr_{S^{n + 1} (V)} ○ d & = π_{n + 1} ○ \pr_{V}^{\otimes (n + 1)} \\ ○ \sum_{k = 0}^{n} (f^{\otimes k} \otimes d \otimes f^{\otimes (n - k)}) Δ_{D}^{n} \\ = π_{n + 1} ○ \sum_{k = 0}^{n} ({(\pr_{V} ○ f)}^{\otimes k} \otimes (\pr_{V} ○ d) \\ \otimes {(\pr_{V} ○ f)}^{\otimes (n - k)}) Δ_{D}^{n} . \end{array}$ $Mathematical equation: \[ \begin{array}{ll} \displaystyle\pr_{\Sy^{n+1}(V)}\circ d &\displaystyle=\pi_{n+1}\circ \pr_V^{\otimes (n+1)}\\ &\displaystyle\quad\circ\sum_{k=0}^n (f^{\otimes k}\otimes d\otimes f^{\otimes(n-k)})\Delta_D^n\\ &\displaystyle=\pi_{n+1}\circ \sum_{k=0}^n ((\pr_V\circ f)^{\otimes k}\otimes (\pr_V\circ d)\\ &\displaystyle\quad\otimes (\pr_V\circ f)^{\otimes(n-k)})\Delta_D^n. \end{array} \]$

As this holds for all $n \in ℕ$ $Mathematical equation: $n\in \N$$ and as $\pr_{k} ○ d = 0$ $Mathematical equation: $\pr_{\mathbb k}\circ d=0$$ by the same computation as in the proof of Proposition 20, d is completely determined by pr_V ○ d.

What is left is to show that given λ ∈ Hom(D, V ) homogeneous, d := μ_S(λ ⊗ f)Δ_D is a coderivation with pr_V○d = λ. While the latter holds by construction, we compute for x ∈ D homogeneous $\begin{array}{l} (Δ_{S} ○ d) (x) & = \sum Δ_{S} (λ (x_{(1)})) \lor Δ_{S} (f (x_{(2)})) \\ = \sum (λ (x_{(1)}) \otimes 1 + 1 \otimes λ (x_{(1)})) \lor (f \otimes f) (Δ_{D} (x_{(2)})) \\ = \sum (λ (x_{(1)}) \otimes 1 + 1 \otimes λ (x_{(1)})) (f (x_{(2)}) \otimes f (x_{(3)})) \\ = \sum (λ (x_{(1)}) \lor f (x_{(2)})) \otimes f (x_{(3)}) + {(- 1)}^{(| λ | + | x_{(1)} |) | x_{(2)} |} f (x_{(2)}) \otimes (λ (x_{(1)}) \lor f (x_{(3)})) \\ = \sum (μ_{S} (λ \otimes f) \otimes f + f \otimes μ_{S} (λ \otimes f)) (x_{(1)} \otimes x_{(2)} \otimes x_{(3)}) \\ = (μ_{S} (λ \otimes f) Δ_{D} \otimes f + f \otimes μ_{S} (λ \otimes f) Δ_{D}) (Δ_{D} (x)), \end{array}$ $Mathematical equation: \[ \begin{array}{ll} \displaystyle(\Delta_S\circ d)(x)&\displaystyle=\sum \Delta_S(\lambda(x_{(1)}))\vee \Delta_S(f(x_{(2)}))\\[6pt] &\displaystyle=\sum \big(\lambda(x_{(1)})\otimes 1+ 1\otimes \lambda(x_{(1)})\big)\vee (f\otimes f)(\Delta_D(x_{(2)}))\\[6pt] &\displaystyle=\sum\big(\lambda(x_{(1)})\otimes 1+1\otimes \lambda(x_{(1)})\big) (f(x_{(2)})\otimes f(x_{(3)}))\\[6pt] &\displaystyle=\sum\big(\lambda(x_{(1)})\vee f( x_{(2)})\big)\otimes f(x_{(3)}) + (-1)^{(|\lambda|+|x_{(1)}|)|x_{(2)}|}f(x_{(2)})\otimes \big(\lambda(x_{(1)})\vee f(x_{(3)})\big)\\[6pt] \displaystyle&\displaystyle=\sum\big(\mu_S (\lambda\otimes f)\otimes f + f\otimes \mu_S(\lambda\otimes f)\big)(x_{(1)}\otimes x_{(2)}\otimes x_{(3)})\\[6pt] &\displaystyle=\big(\mu_S (\lambda\otimes f) \Delta_D\otimes f + f\otimes \mu_S(\lambda\otimes f)\Delta_D\big)(\Delta_D(x)), \end{array} \]$

where we used Lemma 24 in the first and cocommutativity of D in the fifth equality.

2.5 Dual spaces

The graded vector space $V^{*} ≔ Hom (V, k)$ $Mathematical equation: $V^*\vcentcolon=\Hom(V,{\mathbb k})$$ is called the dual space of V. By degree reasons, ${(V^{*})}_{k} = {Hom}_{k} (V, k) = Hom (V_{- k}, k) = {(V_{- k})}^{*}$ $Mathematical equation: $(V^*)_k=\Hom_k(V,{\mathbb k})=\Hom(V_{-k},{\mathbb k})=(V_{-k})^*$$ . For f ∈ Hom_p(V, W), the linear map f^*∈ Hom_p(W^*, V ^*) is defined by $f^{*} (φ) = {(- 1)}^{| φ | | f |} φ ○ f$ $Mathematical equation: $f^*(\varphi)=(-1)^{|\varphi||f|}\varphi\circ f$$ for φ ∈ W^* homogeneous. Note that ${id}_{V}^{*} = \id_{V^{*}}$ $Mathematical equation: $\id_V^*= \id_{V^*}$$ and if g is a homogeneous linear map with domain W, we have ${(g ○ f)}^{*} = {(- 1)}^{| f | | g |} f^{*} ○ g^{*}$ $Mathematical equation: $(g\circ f)^*=(-1)^{|f||g|}f^*\circ g^*$$ .

We say that V is of finite type if V _k is finite-dimensional for all $k \in ℤ$ $Mathematical equation: $k\in\Z$$ . Note that if V is of finite type, the canonical inclusion V ↪V ^** is an isomorphism.

If V _k = 0 for k > 0, then V is called $ℤ_{\leq 0}$ $Mathematical equation: $\Z_{\leq 0}$$ -graded. Notions as $ℤ_{< 0}$ $Mathematical equation: $\Z_{<0}$$ -graded or $ℤ_{\geq 0}$ $Mathematical equation: $\Z_{\geq 0}$$ -graded are defined accordingly. In the following, we denote V ^⊗n as Tⁿ(V ) for better readability.

Proposition 25. If V is of finite type and if for all $k \in ℤ$ $Mathematical equation: $k\in\Z$$ the decomposition $T^{n} {(V)}_{k} = \underset{i_{1} + … + i_{n} = k}{\oplus} (V_{i_{1}} \otimes … \otimes V_{i_{n}})$ $Mathematical equation: \[T^n(V)_{k}=\bigoplus_{i_1+\dotsc+ i_n=k} (V_{i_1}\otimes\dotsc\otimes V_{i_n})\]$ has only finitely many non-trivial summands, then the canonical inclusion $T^{n} (V^{*}) � T^{n} {(V)}^{*}$ $Mathematical equation: $T^n(V^*)\hookrightarrow T^n(V)^*$$ is an isomorphism.

Proof. It is well-known that for finite-dimensional (ungraded) vector spaces V ₁ , …, V _n, the canonical inclusion $V_{1}^{*} \otimes … \otimes V_{n}^{*} � {(V_{1} \otimes … \otimes V_{n})}^{*}$ $Mathematical equation: $V_1^*\otimes\dotsc\otimes V_n^*\hookrightarrow (V_1\otimes\dotsc\otimes V_n)^*$$ is an isomorphism. We then have $\begin{array}{l} {(T^{n} {(V)}_{- k})}^{*} & = & \underset{i_{1} + … + i_{n} = k}{\oplus} {(V_{- i_{1}} \otimes … \otimes V_{- i_{n}})}^{*} \\ ≅ & \underset{i_{1} + … + i_{n} = k}{\oplus} {(V^{*})}_{i_{1}} \otimes … \otimes {(V^{*})}_{i_{n}} \\ = & T^{n} {(V^{*})}_{k} . \end{array}$ $Mathematical equation: \begin{eqnarray*} (T^n(V)_{-k})^*&=&\bigoplus_{i_1+\dotsc+i_n=k}(V_{-i_1}\otimes\dotsc\otimes V_{-i_n})^*\nonumber\\ &\cong&\bigoplus_{i_1+\dotsc+i_n=k}(V^*)_{i_1}\otimes\dotsc\otimes (V^*)_{i_n}\nonumber\\ &=& T^n(V^*)_k. \nonumber \end{eqnarray*}$

Remark 26. If V is of finite type and $ℤ_{\leq 0}$ $Mathematical equation: $\Z_{\leq 0}$$ -graded, V ^* is also of finite type and $ℤ_{\geq 0}$ $Mathematical equation: $\Z_{\geq 0}$$ -graded and they both satisfy the hypothesis of Proposition 25. It is then easy to see that Δ: V → V ⊗ V makes V into a graded coassociative/cocommutative coalgebra if and only if $Δ^{*} : V^{*} \otimes V^{*} ≅ {(V \otimes V)}^{*} \to V^{*}$ $Mathematical equation: $\Delta^*\colon V^*\otimes V^*\cong (V\otimes V)^*\to V^*$$ makes V ^* into an associative/commutative algebra. A linear map d: V → V is then a coderivation of (V, Δ) if and only if − d^* is a derivation of (V ^*, Δ^*). The map 𝔤𝔩(V ) → 𝔤𝔩(V ^*), f↦ − f^* preserves the graded commutator and therefore restricts to an isomorphism of graded Lie algebras Coder(V )≅Der(V ^*).

Corollary 27. If V is of finite type and $ℤ_{< 0}$ $Mathematical equation: $\Z_{< 0}$$ -graded, the canonical inclusion $T (V^{*}) � T {(V)}^{*}$ $Mathematical equation: $T(V^*)\hookrightarrow T(V)^*$$ is an isomorphism.

Proof. Note that for all $k \in ℤ$ $Mathematical equation: $k\in\Z$$ and n > k, we have $T^{n} {(V^{*})}_{- k} = 0$ $Mathematical equation: $T^n(V^*)_{-k}=0$$ . Then $\begin{array}{l} {(T {(V)}_{- k})}^{*} & ≅ & \underset{n \geq 0}{\oplus} {(T^{n} {(V)}_{- k})}^{*} = \underset{n \geq 0}{\oplus} {(T^{n} {(V)}^{*})}_{k} \\ ≅ & \underset{n \geq 0}{\oplus} T^{n} {(V^{*})}_{k} = T {(V^{*})}_{k} . \end{array}$ $Mathematical equation: \begin{eqnarray*} (T (V)_{-k})^*&\cong& \bigoplus_{n\geq 0} (T^n(V)_{-k})^*=\bigoplus_{n\geq 0} (T^n(V)^*)_k \nonumber\\ &\cong& \bigoplus_{n\geq 0} T^n(V^*)_k= T(V^*)_k. \nonumber \end{eqnarray*}$

Lemma 28. Let $ξ \in T^{n} (V^{*}) \subset T^{n} {(V)}^{*}$ $Mathematical equation: $\xi\in T^n(V^*)\subset T^n(V)^*$$ and σ ∈ 𝔖_n. Then $nn\hatsilon (σ) ξ = ξ ○ \hat{ε} (σ^{- 1}) .$ $Mathematical equation: \begin{equation*} \hatsilon(\sigma)\xi=\xi\circ\hatsilon(\sigma^{-1}). \end{equation*}$ (26)

Proof. It suffices to show this for σ = s₁, …, s_n−1, in which case it is an easy computation.

Proposition 29. If V is of finite type and $ℤ_{< 0}$ $Mathematical equation: $\Z_{<0}$$ -graded, $S {(V)}^{*} ≅ S (V^{*})$ $Mathematical equation: $\Sy(V)^*\cong \Sy(V^*)$$ . Under this identification, $Δ_{S}^{*}$ $Mathematical equation: $\Delta_S^*$$ is the usual multiplication on $S (V^{*})$ $Mathematical equation: $\Sy(V^*)$$ .

Proof. Fix n ≥ 0. By Lemma 28, the isomorphism $T^{n} (V^{*}) ≅ T^{n} {(V)}^{*}$ $Mathematical equation: $T^n(V^*)\cong T^n(V)^*$$ maps the subspace of symmetric elements in Tⁿ(V ^*) onto the space of symmetric linear maps $V^{\otimes n} \to k$ $Mathematical equation: ${V^{\otimes n}}\to {\mathbb k}$$ . While the latter is isomorphic to $Hom (S^{n} (V), k) ≅ S^{n} {(V)}^{*}$ $Mathematical equation: $\Hom(\Sy^n(V),{\mathbb k})\cong \Sy^n(V)^*$$ by Remark 2, the former is isomorphic to $S^{n} (V^{*})$ $Mathematical equation: $\Sy^n(V^*)$$ via the linear map $\begin{array}{l} S^{n} (V^{*}) \to T^{n} (V^{*}), \\ v_{1} \lor … \lor v_{n} \mapsto \sum_{σ \in S_{n}} ε (σ) v_{σ (1)} \otimes … \otimes v_{σ (n)} . \end{array}$ $Mathematical equation: \begin{eqnarray*} &&\Sy^n(V^*)\to T^n(V^*),\nonumber\\ && v_1\vee\dotsc\vee v_n\mapsto \sum_{\sigma\in\mathfrak S_n}\varepsilon(\sigma)v_{\sigma(1)}\otimes\dotsc\otimes v_{\sigma(n)}.\nonumber \end{eqnarray*}$

This yields an isomorphism $S^{n} (V^{*}) ≅ S^{n} {(V)}^{*}$ $Mathematical equation: $\Sy^n(V^*)\cong \Sy^n(V)^*$$ . With a similar reasoning as in the proof of Corollary 27, one obtains $S (V^{*}) ≅ S {(V)}^{*}$ $Mathematical equation: $\Sy(V^*)\cong\Sy(V)^*$$ . It is then a straightforward computation to show that $Δ_{S}^{*}$ $Mathematical equation: $\Delta_S^*$$ is indeed the usual multiplication on $S (V^{*})$ $Mathematical equation: $\Sy(V^*)$$ .

3 L_∞-algebras

We start this section with a theorem from [5] that characterises certain L_∞ -algebras using Lie algebra cohomology; later, we seek to generalise it in the context of L_∞ -algebra cohomology. After that, we discuss different characterisations of L_∞-structures using the key results of Section 2. Different points of view naturally lead to different notions of homomorphisms between L_∞ -algebras; we will finish the section with a comparison of those. For this, we will mostly follow [3], although the original references for Section 3.2 are [1] and [8].

Definition 30. An L_∞-algebra is a graded vector space L together with antisymmetric linear maps l_k: L^⊗k → L called (higher) brackets of degree |l_k| = 2 − k for 1 ≤ k < ∞ such that the generalized Jacobi identity $\begin{array}{l} \sum_{i + j = n + 1} \sum_{σ \in \Sh_{i, n - i}^{- 1}} {(- 1)}^{i (j - 1)} χ (σ) \\ \cdot l_{j} (l_{i} (x_{1}, …, x_{i}), x_{i + 1}, …, x_{n}) = 0 \end{array}$ $Mathematical equation: \begin{eqnarray*}&&\sum_{i+j=n+1}\sum_{\sigma\in\Sh^{-1}_{i,n-i}}(-1)^{i(j-1)}\chi(\sigma) \nonumber\\ &&\quad\cdot l_j(l_i(x_1,\dotsc,x_i),x_{i+1},\dotsc,x_n)=0 \end{eqnarray*}$ (27)

holds for all n ≥ 1 and x₁, …, x_n ∈ L homogeneous. We then callthe set {l_k∣1 ≤ k < ∞} an L_∞-structure on L.

Writing out (27) for n = 1, 2, 3 yields $\begin{array}{l} l_{1} (l_{1} (x_{1})) & = & 0, \\ l_{1} (l_{2} (x_{1}, x_{2})) & = & l_{2} (l_{1} (x_{1}), x_{2}) + {(- 1)}^{| x_{1} |} l_{2} (x_{1}, l_{1} (x_{2})), \\ 0 & = & \sum_{σ \in \Sh_{2, 1}^{- 1}} χ (σ) l_{2} (l_{2} (x_{σ (1)}, x_{σ (2)}), x_{σ (3)}) + l_{1} (l_{3} (x_{1}, x_{2}, x_{3})) + l_{3} (l_{1} (x_{1}), x_{2}, x_{3}) \\ [- 12 p t] & + {(- 1)}^{| x_{1} |} l_{3} (x_{1}, l_{1} (x_{2}), x_{3}) + {(- 1)}^{| x_{1} | + | x_{2} |} l_{3} (x_{1}, x_{2}, l_{1} (x_{3})) \end{array}$ $Mathematical equation: \begin{eqnarray*} l_1(l_1(x_1)) &=&0,\nonumber\\ l_1(l_2(x_1,x_2))&=& l_2(l_1(x_1),x_2)+(-1)^{|x_1|}l_2(x_1,l_1(x_2)),\nonumber\\ 0&=&\sum_{\sigma\in\Sh^{-1}_{2,1}}\chi(\sigma)l_2(l_2(x_{\sigma(1)},x_{\sigma(2)}),x_{\sigma(3)})+ l_1(l_3(x_1,x_2,x_3))+ l_3(l_1(x_1),x_2,x_3) \nonumber\\[-12pt] &&\hspace*{65pt}+ (-1)^{|x_1|}l_3(x_1,l_1(x_2),x_3)+ (-1)^{|x_1|+|x_2|}l_3(x_1,x_2,l_1(x_3))\nonumber \end{eqnarray*}$

for all x₁, x₂, x₃ ∈ L homogeneous. While the first two equations may be summarized by saying that l₁ is a differential on the (non-associative) graded algebra (L, l₂), a comparison with (11) shows that the third one describes the defect of the Jacobi identity in (L, l₂). In particular, an L_∞-algebra with l_k = 0 for k ≥ 3 is nothing else than a DGLA.

If L is concentrated in degree zero, l_k = 0 for k≠2 by degree reasons and (L, l₂) is an (ungraded) Lie algebra.

Definition 31. Let L and L′ be L_∞ -algebras with L_∞ -structures ${l_{k}}_{k \in ℕ}$ $Mathematical equation: $\{l_k\}_{k\in\N}$$ and ${l_{k^{'}}}_{k \in ℕ}$ $Mathematical equation: $\{l_k'\}_{k\in\N}$$ , respectively. A strict L∞ -algebra homomorphism is a degree preserving linear map f : L → L′ satisfying $f ○ l_{k} = l_{l^{'}} ○ f^{\otimes k}$ $Mathematical equation: \begin{equation*} f\circ l_k = l_l'\circ f^{\otimes k} \end{equation*}$ (28)

for all 1 ≤ k < ∞.

These homomorphisms are strict in the sense that they strictly preserve all brackets. A different characterisation of L_∞ -algebras will later lead to a more general notion of L_∞-algebra homomorphisms.

3.1 Characterisation via Lie algebra cohomology

For an (ungraded) Lie algebra (𝔤, [⋅, ⋅]), a representation of 𝔤 on an (ungraded) vector space V is a homomorphism of Lie algebras ρ: 𝔤 → 𝔤𝔩(V ). Given such a ρ, the Lie algebra cohomology with values in V is the cohomology of the Chevalley–Eilenberg (cochain) complex $(\underset{n \geq 0}{\oplus} Hom (\Asy^{n} g, V), δ)$ $Mathematical equation: $\big(\bigoplus_{n\geq 0}\Hom(\Asy^n \mathfrak g, V), \delta\big)$$ , where for an antisymmetric linear map ω: 𝔤^⊗n → V , we define δω by $\begin{array}{l} δ ω (x_{1}, …, x_{n + 1}) \\ = \begin{array}{l} \sum_{i = 1}^{n + 1} {(- 1)}^{i + 1} ρ (x_{i}) (ω (x_{1}, …, {\hat{x}}_{i}, …, x_{n + 1})) \\ + \sum_{1 \leq j < k \leq n + 1} {(- 1)}^{j + k} \\ \cdot ω ([x_{j}, x_{k}], x_{1}, …, {\hat{x}}_{j}, …, {\hat{x}}_{k}, …, x_{n + 1}), \end{array} \end{array}$ $Mathematical equation: \begin{eqnarray*} &&\displaystyle\delta\omega(x_1,\dotsc, x_{n+1})\nonumber\\ &&\displaystyle\quad=\begin{array}{ll}&\displaystyle\sum_{i=1}^{n+1}(-1)^{i+1} \rho(x_i)(\omega(x_1,\dotsc,\hat x_i,\dotsc,x_{n+1})) \nonumber\\ &\displaystyle+\sum_{1\leq j<k\leq n+1}(-1)^{j+k}\nonumber\\ &\displaystyle\cdot\omega([x_j,x_k],x_1,\dotsc,\hat x_j,\dotsc,\hat x_k,\dotsc,x_{n+1}),\end{array}\nonumber \end{eqnarray*}$

for x₁, …, x_n+1 ∈ 𝔤. In the sums above, elements with ̂ are to be omitted.

Theorem 32 ([5], Theorem 55). There is for each n ≥ 1 a one-to-one correspondence between L_∞-algebras L such that L_k = 0 for k≠ − n, 0 and l₁ = 0 and quadruples (𝔤, V, ρ, l_n+2) consisting of a Lie algebra 𝔤, a representation ρ of 𝔤 on a vector space V and an (n + 2)-cocycle l_n+2.

Sketch of proof. For an L_∞-algebra L = L₀ ⊕ L_−n with l₁ = 0, all brackets exept for l₂ and l_n+2 have to vanish by degree reasons. Also, l₂ has to vanish on ∧²L_−n and l_n+2 can only be non-trivial on ∧ⁿ⁺²L₀ with image in L_−n. Using (7), we can decompose l₂ into linear maps [⋅, ⋅]: ∧²L₀ → L₀ and ρ: L₀ ⊗ L_−n → L_−n. It is then a matter of computation to show that l₂ and l_n+2 satisfying (27) amounts to (L₀, [⋅, ⋅]) being a Lie algebra, ρ being a representation of L₀ on L_−n and l_n+2 being a cocycle.

3.2 Symmetric brackets and codifferentials

Recall that by Corollary 4, an antisymmetric map l_k : L^⊗k → L of degree 2 − k is equivalent to a symmetric degree one map $λ_{k} : S^{k} (L [1]) \to L [1]$ $Mathematical equation: $\lambda_k\colon \Sy^k(L[1])\to L[1]$$ such that $l_{k} = ↑ ○ λ_{k} ○ ↓^{\otimes k} .$ $Mathematical equation: \[ l_k = {\uparrow}\circ \lambda_k \circ {\downarrow^{\otimes k}}. \]$

If we now rewrite (27) in terms of the maps λ_k , we obtain a characterisation of L_∞-structures on L in terms of symmetric brackets. Note that for fixed n, we can write (27) as $\sum_{i + j = n + 1} \sum_{σ \in \Sh_{i, n - i}^{- 1}} {(- 1)}^{(j - 1) i} l_{j} (l_{i} \otimes \id_{L}^{\otimes (j - 1)}) \hat{χ} (σ) = 0$ $Mathematical equation: \[ \sum_{i+j=n+1} \sum_{\sigma \in \Sh^{-1}_{i,n-i}}(-1)^{(j-1)i} l_j( l_i \otimes \id_L^{\otimes(j-1)}) \hat\chi(\sigma)=0 \]$

As ↓ and ↑^⊗n are isomorphisms, this is equivalent to $\begin{matrix} 0 & = {(- 1)}^{\frac{n (n - 1)}{2}} \sum_{i + j = n + 1} \sum_{σ \in \Sh_{i, n - i}^{- 1}} {(- 1)}^{(j - 1) i} ↓ ○ l_{j} (l_{i} \otimes \id_{L}^{\otimes (j - 1)}) \hat{χ} (σ) ○ ↑^{\otimes n} \\ = {(- 1)}^{\frac{n (n - 1)}{2}} \sum_{i + j = n + 1} \sum_{σ \in \Sh_{i, n - i}^{- 1}} {(- 1)}^{(j - 1) i} λ_{j} ○ ↓^{\otimes j} (l_{i} \otimes \id_{L}^{\otimes (j - 1)}) ↑^{\otimes n} ○ \hat{ε} (σ) \\ = {(- 1)}^{\frac{n (n - 1)}{2}} \sum_{i + j = n + 1} \sum_{σ \in \Sh_{i, n - i}^{- 1}} λ_{j} ((↓ ○ l_{i}) \otimes ↓^{\otimes (j - 1)}) ↑^{\otimes n} ○ \hat{ε} (σ) \\ = \sum_{i + j = n + 1} \sum_{σ \in \Sh_{i, n - i}^{- 1}} λ_{j} (λ_{i} \otimes \id_{L [1]}^{\otimes (j - 1)}) \hat{ε} (σ), \end{matrix}$ $Mathematical equation: \[\begin{array}{@{}l@{}l@{}} 0&\displaystyle=(-1)^{\frac{n(n-1)}{2}}\sum_{i+j=n+1} \sum_{\sigma \in \Sh^{-1}_{i,n-i}}(-1)^{(j-1)i}{\downarrow}\circ l_j( l_i \otimes \id_L^{\otimes(j-1)}) \hat\chi(\sigma)\circ{\uparrow^{\otimes n}}\\ &\displaystyle=(-1)^{\frac{n(n-1)}{2}}\sum_{i+j=n+1} \sum_{\sigma \in \Sh^{-1}_{i,n-i}}(-1)^{(j-1)i}\lambda_j\circ {\downarrow^{\otimes j}}( l_i \otimes \id_L^{\otimes(j-1)}){\uparrow^{\otimes n}}\circ \hatsilon(\sigma)\\ &\displaystyle=(-1)^{\frac{n(n-1)}{2}}\sum_{i+j=n+1} \sum_{\sigma \in \Sh^{-1}_{i,n-i}}\lambda_j( ({\downarrow}\circ l_i) \otimes {\downarrow^{\otimes(j-1)}}){\uparrow^{\otimes n}}\circ \hatsilon(\sigma)\\ &\displaystyle=\sum_{i+j=n+1} \sum_{\sigma \in \Sh^{-1}_{i,n-i}}\lambda_j( \lambda_i \otimes \id_{L[1]}^{\otimes(j-1)})\hatsilon(\sigma), \end{array} \]$

where we used (2) and (4). We have proved the following.

Proposition 33. An L_∞-structure {l_k ∣1 ≤ k < ∞} on the graded vector space L is equivalent to a system of linear maps $λ_{k} : S^{k} (L [1]) \to L [1]$ $Mathematical equation: $\lambda_k \colon \Sy^k(L[1])\to L[1]$$ for 1 ≤ k < ∞, all of degree one, such that $\begin{array}{l} \sum_{i + j = n + 1} \sum_{σ \in \Sh_{i, n - i}^{- 1}} ε (σ) \\ \cdot λ_{j} (λ_{i} (x_{σ (1)}, …, x_{σ (i)}), x_{σ (i + 1)}, …, x_{σ (n)}) = 0 \end{array}$ $Mathematical equation: \begin{eqnarray*}&&\sum_{i+j=n+1} \sum_{\sigma \in \Sh^{-1}_{i,n-i}}\varepsilon(\sigma)\nonumber\\ &&\quad\cdot\lambda_j(\lambda_i(x_{\sigma(1)},\dotsc,x_{\sigma(i)}),x_{\sigma(i+1)},\dotsc,x_{\sigma(n)})=0\nonumber\\ \end{eqnarray*}$ (29) holds for all for all n ≥ 1 and x₁, …, x_n ∈ L[1] homogeneous.

Corollary 34. An L_∞-structure on the graded vector space L is equivalent to a linear degree one map $λ : \rsym (L [1]) \to L [1]$ $Mathematical equation: $\lambda \colon\rsym (L[1])\to L[1]$$ such that $λ ○ μ_{S} (λ \otimes \id_{S}) Δ_{S} = 0.$ $Mathematical equation: \begin{equation*}\lambda\circ \mu_S(\lambda\otimes \id_S)\Delta_S=0. \end{equation*}$ (30)

Proof. Combine the brackets in Proposition 33 to a single element $λ = \sum_{k} λ_{k} \in Hom (\rsym (L [1]), L [1])$ $Mathematical equation: $\lambda=\sum_k \lambda_k\in\Hom(\rsym(L[1]),L[1])$$ of degree one and compare with (18).

By abuse of notation, we then also refer to the pair (L[1], λ) as an L_∞-algebra. Strict homomorphisms of L_∞-algebras can then be described as degree preserving linear maps that preserve the symmetric brackets.

Proposition 35. Let (L[1], λ) and (L′[1], λ′) be L_∞-algebras. There is a one-to-one correspondence between strict L_∞-algebra homomorphisms f : L → L′ and linear degree preserving maps g =↓○ f○ ↑: L[1] → L′[1] satisfying $g ○ λ_{k} = λ_{k^{'}} ○ g^{\otimes k}$ $Mathematical equation: \begin{equation*}g\circ \lambda_k = \lambda_k'\circ g^{\otimes k} \end{equation*}$ (31) for all k ≥ 1.

The equation (31) can be written more conveniently as $g ○ λ = λ^{'} ○ S (g) .$ $Mathematical equation: \begin{equation*}g\circ \lambda = \lambda'\circ \Sy(g). \end{equation*}$ (32)

We then also refer to g as a strict L_∞-algebra homomorphism.

Theorem 36. An L_∞-structure on the graded vector space L is equivalent to a codifferential $d \in \Coder (S (L [1]))$ $Mathematical equation: $d\in\Coder(\Sy(L[1]))$$ with d(1) = 0.

In this case, we also refer to the pair $(S (L [1]), d)$ $Mathematical equation: $(\Sy(L[1]),d)$$ as an L_∞-algebra.

Proof. Let $λ : \rsym (L [1]) \to L [1]$ $Mathematical equation: $\lambda \colon\rsym(L[1])\to L[1]$$ be of degree one and d = μ_S(λ ⊗id_S)Δ_S be the unique coderivation extending λ in the sense of Theorem 22. As $d^{2} = \frac{1}{2} [d, d]$ $Mathematical equation: $d^2=\frac{1}{2}[d,d]$$ is a coderivation of $S (L [1])$ $Mathematical equation: $\Sy(L[1])$$ by Proposition 21, we have by Theorem 22 that d² = 0 if and only if $0 = \pr_{L [1]} ○ d^{2} = λ ○ μ_{S} (λ \otimes \id_{S}) Δ_{S} .$ $Mathematical equation: \[ 0=\pr_{L[1]}\circ d^2= \lambda \circ \mu_S(\lambda\otimes \id_S)\Delta_S. \]$

Corollary 37. If L is of finite type and $ℤ_{\leq 0}$ $Mathematical equation: $\Z_{\leq 0}$$ -graded, an L_∞-structure on L is equivalent to a differential on the graded algebra $S (L {[1]}^{*})$ $Mathematical equation: $\Sy(L[1]^*)$$ . Explicitly, consider d_CE = −d^* for d as in Theorem 36.

Proof. See Remark 26 and note that as L is $ℤ_{\leq 0}$ $Mathematical equation: $\Z_{\leq 0}$$ -graded, each $d \in \Coder (S (L [1]))$ $Mathematical equation: $d\in\Coder(\Sy(L[1]))$$ of degree one vanishes on $k$ $Mathematical equation: ${\mathbb k}$$ by degree reasons.

3.3 Weak homomorphisms

Let $(S (L [1]), d)$ $Mathematical equation: $(\Sy(L[1]),d)$$ and $(S (L^{'} [1]), d^{'})$ $Mathematical equation: $(\Sy(L'[1]),d')$$ be L_∞ -algebras, λ = pr_L[1]○d and $λ^{'} = \pr_{L^{'} [1]} ○ d^{'}$ $Mathematical equation: $\lambda'=\pr_{L'[1]}\circ d'$$ . The characterisation of L_∞-structures as codifferentials on the symmetric coalgebra leads to another notion of homomorphisms of L_∞ -algebras, namely as homomorphisms of (coaugmented) DGCs.

Definition 38. A (weak) homomorphism of L∞-algebras between L and L′ is a homomorphism of coaugmented DGCs $f : S (L [1]) \to S (L^{'} [1])$ $Mathematical equation: $f\colon \Sy(L[1])\to \Sy(L'[1])$$ .

Remark 39. By Proposition 21 and Theorem 22, a homomorphism of coaugmented coalgebras $f : S (L [1]) \to S (L^{'} [1])$ $Mathematical equation: $f\colon \Sy(L[1])\to \Sy(L'[1])$$ is a homomorphism of L_∞-algebras if and only if $(\pr_{L^{'} [1]} ○ f) ○ d = λ^{'} ○ f .$ $Mathematical equation: \[(\pr_{L'[1]}\circ f)\circ d=\lambda'\circ f.\]$

Note that by Proposition 20, it makes sense to also refer to DGC homomorphisms $\rsym (L [1]) \to \rsym (L^{'} [1])$ $Mathematical equation: $\rsym(L[1])\to \rsym(L'[1])$$ as homomorphisms of L_∞-algebras.

From the dualised standpoint, we immediately get the following.

Proposition 40. Assume that L and L′ are $ℤ_{\leq 0}$ $Mathematical equation: $\Z_{\leq 0}$$ -graded and of finite type. Then $f : S (L [1]) \to S (L^{'} [1])$ $Mathematical equation: $f\colon \Sy(L[1])\to \Sy(L'[1])$$ is a homomorphism of L_∞-algebras if and only if $f^{*} : S (L^{'} {[1]}^{*}) \to S (L {[1]}^{*})$ $Mathematical equation: $f^*\colon \Sy(L'[1]^*)\to \Sy(L[1]^*)$$ is a homomorphism of unital DG algebras.

With now two different notions of L_∞-algebra homomorphisms at hand, it is reasonable to ask if there is a connection between them. As commented in ([8], Remark 5.3), strict homomorphisms are essentially the weak homomorphisms that preserve the exterior degree.

Lemma 41. Let g: L[1] → L′[1] be a linear degree preserving map. Then g is a strict L_∞-algebra homomorphism if and only if $S (g)$ $Mathematical equation: $\Sy(g)$$ is a weak one.

Proof. Observe that $\pr_{L^{'} [1]} ○ S (g) ○ d = g ○ \pr_{L [1]} ○ d = g ○ λ$ $Mathematical equation: $\pr_{L'[1]}\circ \Sy(g)\circ d = g\circ \pr_{L[1]}\circ d= g\circ \lambda$$ .

Lemma 42. A homomorphism of coalgebras $f : S (L [1]) \to S (L^{'} [1])$ $Mathematical equation: $f\colon \Sy(L[1])\to \Sy(L'[1])$$ preserves the exterior degree if and only if $f = S (g)$ $Mathematical equation: $f=\Sy(g)$$ for a linear degree preserving map g: L[1] → L′[1].

Proof. Assume that $f : S (L [1]) \to S (L^{'} [1])$ $Mathematical equation: $f\colon \Sy(L[1])\to \Sy(L'[1])$$ is a homomorphism of coalgebras such that $f (S^{n} (L [1])) \subset S^{n} (L^{'} [1])$ $Mathematical equation: $f(\Sy^n(L[1]))\subset \Sy^n(L'[1])$$ for all n and let $g ≔ \pr_{L^{'} [1]} ○ {f |}_{L [1]}$ $Mathematical equation: $g\vcentcolon = \pr_{L'[1]}\circ \left. f \right|_{L[1]}$$ . Then $\pr_{L^{'} [1]} ○ f = g ○ \pr_{L [1]} = \pr_{L^{'} [1]} ○ S (g)$ $Mathematical equation: $\pr_{L'[1]}\circ f=g\circ \pr_{L[1]}=\pr_{L'[1]}\circ \Sy(g)$$ . Hence, $f = S (g)$ $Mathematical equation: $f=\Sy(g)$$ by Proposition 18.

Proposition 43. Let $f : S (L [1]) \to S (L^{'} [1])$ $Mathematical equation: $f\colon \Sy(L[1])\to \Sy(L'[1])$$ be a (weak) L_∞-algebra homomorphism. Then f preserves the exterior degree if and only if $f = S (g)$ $Mathematical equation: $f=\Sy(g)$$ for a strict L_∞-algebra homomorphism g.

Proof. Combine Lemma 41 and Lemma 42.

From this it follows for example that all (weak) L_∞ -algebra homomorphisms between Lie algebras are induced by Lie algebra homomorphisms.

4 Representations (up to homotopy)

While representations (up to homotopy) of L_∞-algebras are often defined in terms of antisymmetric maps, we start with a definition that keeps the symmetric point of view of the last section. While it is a straightforward computation to show equivalence between these definitions, it is convenient to save this for Section 5.1. We then show that representations (up to homotopy) are nothing else than weak L_∞ -algebra homomorphisms into 𝔤𝔩(V ) for a DG vector space V , a characterisation due to Lada and Markl [8]. In [3], representations (up to homotopy) were described (under some finiteness assumptions) as differentials on $S (L {[1]}^{*}) \otimes V$ $Mathematical equation: $\Sy(L[1]^*)\otimes V$$ . We discuss this point of view in the second half of this section, which also leads us to L_∞ -algebra cohomology.

From now on, L denotes an L_∞-algebra with L_∞-structure {l_k ∣ 1 ≤ k < ∞} and λ and d are as in Corollary 34 and Theorem 36, respectively.

Definition 44. A representation (up to homotopy) of L on V is a linear map $ρ : S (L [1]) \otimes V \to V$ $Mathematical equation: ${\rho\colon \Sy(L[1])\otimes V\to V}$$ of degree one that satisfies $ρ (d \otimes \id_{V}) + ρ ({id}_{S} \otimes ρ) (Δ_{S} \otimes \id_{V}) = 0.$ $Mathematical equation: \begin{equation*}\rho(d\otimes \id_V)+\rho(\id_S\otimes \rho)(\Delta_S\otimes \id_V)=0. \end{equation*}$ (33)

4.1 Representations as (weak) homomorphisms

We prove the following version of ([8], Theorem 5.2).

Theorem 45. There is a one-to-one correspondence between representations of L on V and pairs $(\partial, \tilde{f})$ $Mathematical equation: $(\partial,\tilde f)$$ , where ∂ is a differential on V and $\tilde{f} : S (L [1]) \to S (g l (V) [1])$ $Mathematical equation: $\tilde f\colon \Sy(L[1])\to \Sy(\mathfrak{gl}(V)[1])$$ a homomorphism of L_∞-algebras. Here, 𝔤𝔩(V ) carries the DGLA structure induced by ∂, see Example 12.

One should therefore really think of L being representated on a DG vector space. The following lemma characterises L_∞ -algebra homomorphisms into DGLAs and is a symmetric version of ([8], Definition 5.2).

Lemma 46. Let $(L^{'}, {l^{'}}_{2}, {l^{'}}_{1})$ Mathematical equation: $(L',l'_2,l'_1)$ be a DGLA and $λ^{'} = {λ^{'}}_{1} + {λ^{'}}_{2}$ $Mathematical equation: $\lambda'=\lambda'_1+\lambda'_2$$ be the corresponding linear degree one map $\rsym (L^{'} [1]) \to L^{'} [1]$ $Mathematical equation: $\rsym(L'[1])\to L'[1]$$ . For $f : \ol\Sy (L [1]) \to L^{'} [1]$ $Mathematical equation: $f\colon \ol\Sy(L[1])\to L'[1]$$ a linear degree preserving map, the induced homomorphism of coalgebras $\tilde{f} : \rsym (L [1]) \to \rsym (L^{'} [1])$ $Mathematical equation: $\tilde f\colon \rsym(L[1])\to \rsym(L'[1])$$ is a homomorphism of L_∞-algebras if and only if $f ○ \ol d = λ_{1^{'}} ○ f + \frac{1}{2} λ_{2^{'}} (f \otimes f) \ol Δ_{S} .$ $Mathematical equation: \begin{equation*}f\circ \ol d= \lambda_1'\circ f + \frac{1}{2}\lambda_2'(f\otimes f)\ol\Delta_S. \end{equation*}$ (34)

This is the case if and only if the linear degree one map $ρ : \rsym (L [1]) \to L^{'}$ $Mathematical equation: $\rho\colon \rsym(L[1])\to L'$$ defined by $f (x) = {(- 1)}^{| x | + 1} ↓ ρ (x)$ $Mathematical equation: $f(x)=(-1)^{|x|+1}\downarrow \rho(x)$$ satisfies $ρ ○ \ol d + {l^{'}}_{1} ○ ρ + \frac{1}{2} {l^{'}}_{2} (ρ \otimes ρ) \ol Δ_{S} = 0.$ $Mathematical equation: \begin{equation*}\rho\circ \ol d +l'_1\circ \rho +\frac{1}{2} l'_2(\rho\otimes \rho)\ol \Delta_S=0. \end{equation*}$ (35)

Proof. The first part follows immediately from Remark 39 and the explicit construction of $\tilde{f}$ $Mathematical equation: $\tilde f$$ (see Proposition 18). It is straightforward to check that for $x \in \rsym (L [1])$ $Mathematical equation: $x\in \rsym(L[1])$$ homogeneous, $\begin{array}{l} f (d (x)) & = & {(- 1)}^{| x |} ↓ ρ (d (x)), \\ {λ^{'}}_{1} (f (x)) & = & {(- 1)}^{| x | + 1} ↓ l_{1^{'}} (ρ (x)), \\ ({λ^{'}}_{2} (f \otimes f) \ol Δ_{S}) (x) & = & {(- 1)}^{| x | + 1} ↓ ({l^{'}}_{2} (ρ \otimes ρ) \ol Δ_{S}) (x), \end{array}$ $Mathematical equation: \begin{eqnarray*} f(d(x))&=&(-1)^{|x|}\downarrow\rho(d(x)),\nonumber\\ \lambda'_1(f(x))&=&(-1)^{|x|+1}\downarrow l_1'(\rho(x)),\nonumber\\ \big(\lambda'_2(f\otimes f)\ol\Delta_S\big)(x)&=&(-1)^{|x|+1}\downarrow \big(l'_2(\rho\otimes\rho)\ol\Delta_S\big)(x),\nonumber \end{eqnarray*}$

from which the second part then follows.

Proof of Theorem 45: As $Hom (S (L [1]) \otimes V, V) ≅ \Hom (S (L [1]), \gl (V))$ $Mathematical equation: $\Hom(\Sy(L[1])\otimes V,V)\cong \Hom(\Sy(L[1]),\gl(V))$$ , a linear degree one map $ρ : S (L [1]) \otimes V \to V$ $Mathematical equation: $\rho\colon \Sy(L[1])\otimes V \to V$$ can be decomposed into linear degree one maps $\tilde{ρ} : \ol\Sy (L [1]) \to \gl (V)$ $Mathematical equation: $\tilde\rho\colon \ol\Sy(L[1])\to \gl(V)$$ and $ρ_{0} : k \to \gl (V)$ $Mathematical equation: $\rho_0\colon {\mathbb k} \to \gl(V)$$ ; the latter being equivalent to the choice of a degree one element $\partial = ρ_{0} (1_{k}) \in \gl (V)$ $Mathematical equation: $\partial=\rho_0(1_{\mathbb k})\in \gl(V)$$ . If we show that under this identification ρ satisfying (33) is equivalent to ∂² = 0 and $\tilde{ρ}$ $Mathematical equation: $\tilde \rho$$ satisfying (35), the assertion follows by Lemma 46. For $x \in \rsym (L [1])$ $Mathematical equation: $x\in \rsym(L[1])$$ homogeneous, $\begin{array}{l} \frac{1}{2} (\bracket (\tilde{ρ} \otimes \tilde{ρ}) \ol Δ_{S}) (x) \\ = \frac{1}{2} \sum {(- 1)}^{| x_{(1)} |} [\tilde{ρ} (x_{(1)}), \tilde{ρ} (x_{(2)})] \\ = \frac{1}{2} \sum {(- 1)}^{| x_{(1)} |} \tilde{ρ} (x_{(1)}) ○ \tilde{ρ} (x_{(2)}) \\ + {(- 1)}^{| x_{(1)} | | x_{(2)} | + | x_{(2)} |} \tilde{ρ} (x_{(2)}) ○ \tilde{ρ} (x_{(1)}) \\ = \frac{1}{2} \sum ρ ({id}_{S} \otimes ρ) (x_{(1)} \otimes x_{(2)} \\ + {(- 1)}^{| x_{(1)} | | x_{(2)} |} x_{(2)} \otimes x_{(1)}, \ea) \\ = ρ ({id}_{S} \otimes ρ) (\ol Δ_{S} (x), \ea) \end{array}$ $Mathematical equation: \[\begin{array}{ll} &\displaystyle\frac{1}{2}\big(\bracket(\tilde\rho\otimes \tilde\rho) \ol\Delta_{S}\big)(x)\\[6pt] &\displaystyle\quad=\frac{1}{2}\sum (-1)^{|x_{(1)}|}[\tilde\rho(x_{(1)}),\tilde\rho(x_{(2)})]\\[6pt] &\displaystyle\quad=\frac{1}{2}\sum (-1)^{|x_{(1)}|}\tilde\rho(x_{(1)})\circ \tilde\rho(x_{(2)})\\[6pt] &\displaystyle\qquad+(-1)^{|x_{(1)}||x_{(2)}|+|x_{(2)}|}\tilde\rho(x_{(2)})\circ\tilde\rho(x_{(1)})\\[6pt] &\displaystyle\quad=\frac{1}{2}\sum \rho(\id_{S}\otimes \rho)(x_{(1)}\otimes x_{(2)}\\[6pt] &\displaystyle\qquad+(-1)^{|x_{(1)}||x_{(2)}|}x_{(2)}\otimes x_{(1)},\ea)\\[6pt] &\displaystyle\quad=\rho(\id_{S}\otimes \rho)(\ol\Delta_{S}(x),\ea) \end{array}\]$

by cocommutativity of $\rsym (L [1])$ $Mathematical equation: $\rsym(L[1])$$ and $\begin{array}{l} [\partial, \tilde{ρ} (x)] & = & \partial ○ \tilde{ρ} (x) - {(- 1)}^{| \tilde{ρ} (x) |} \tilde{ρ} (x) ○ \partial \\ = & ρ_{0} (1) ○ \tilde{ρ} (x) + {(- 1)}^{| x |} \tilde{ρ} (x) ○ ρ_{0} (1) \\ = & ρ ({id}_{S} \otimes ρ) (1 \otimes x + x \otimes 1, \ea) . \end{array}$ $Mathematical equation: \begin{eqnarray*} [\partial,\tilde\rho(x)]&=& \partial\circ \tilde\rho(x) -(-1)^{|\tilde\rho(x)|}\tilde\rho(x)\circ\partial\nonumber\\ &=& \rho_0 (1)\circ \tilde\rho (x) + (-1)^{|x|}\tilde\rho (x)\circ \rho_0(1)\nonumber\\ &=& \rho(\id_S\otimes \rho)(1\otimes x + x\otimes 1,\ea).\nonumber \end{eqnarray*}$

As $(\tilde{ρ} ○ d) (x) = ρ (d \otimes \id_{V}) (x, \ea)$ $Mathematical equation: $(\tilde\rho\circ d)(x) = \rho(d\otimes \id_V)(x,\ea)$$ , $\tilde{ρ}$ $Mathematical equation: $\tilde\rho$$ satisfying (35) is equivalent to (33) holding on $\rsym (L [1]) \otimes V$ $Mathematical equation: $\rsym(L[1])\otimes V$$ . We also have $\begin{array}{l} (ρ (d \otimes \id_{V}) + ρ ({id}_{S} \otimes ρ) (Δ_{S} \otimes \id_{V})) (1, \cdot) \\ = ρ (1, ρ (1, \cdot)) = \partial^{2}, \end{array}$ $Mathematical equation: \[ \begin{array}{lll} &&\big(\rho(d\otimes \id_V)+\rho(\id_S\otimes \rho)(\Delta_S\otimes \id_V)\big)(1,\,\cdot\,)\\ &&\quad=\rho(1,\rho(1,\,\cdot\,))=\partial^2,\\ \end{array} \]$

which completes the proof as $S (L [1]) = k \oplus \rsym (L [1])$ $Mathematical equation: $\Sy(L[1])={\mathbb k}\oplus \rsym(L[1])$$ .

Example 47 (The trivial representation on a DG vector space). Let (V, ∂) be a DG vector space. There is a trivial strict homomorphism of L_∞-algebras 0: L → 𝔤𝔩(V ). The induced representation $S (L [1]) \otimes V \to V$ $Mathematical equation: $\Sy(L[1])\otimes V\to V$$ is on $k \otimes V ≅ V$ $Mathematical equation: ${\mathbb k} \otimes V\cong V$$ given by ∂ and zero elsewhere and is called the trivial representation of L on V . In particular, there is a trivial representation of L on $k$ $Mathematical equation: ${\mathbb k}$$ .

Remark 48. Let ρ be a representation of L on V and $(\partial, \tilde{f})$ $Mathematical equation: $(\partial, \tilde f)$$ as in Theorem 45.

(1)
Then − ∂^* is a differential on V ^* and the map 𝔤𝔩(V ) → 𝔤𝔩(V ^*), g↦ − g^* is a homomorphism of DGLAs. By composing the corresponding weak homomorphism with $\tilde{f}$ $Mathematical equation: $\tilde f$$ , we obtain an L_∞-algebra homomorphism $S (L [1]) \to S (\gl (V^{*}) [1])$ $Mathematical equation: $\Sy(L[1])\to \Sy(\gl(V^*)[1])$$ . The induced representation is given by $ρ^{\lor} : S (L [1]) \otimes V^{*} \to V^{*}, x \otimes ξ \mapsto - ρ {(x, \ea)}^{*} ξ$ $Mathematical equation: \[ \rho^{\vee} \colon \Sy(L[1])\otimes V^* \to V^*, \quad x\otimes \xi \mapsto -\rho(x,\ea)^*\xi \]$ and is called the representation dual to ρ.
(2)
Fix $n \in ℤ$ $Mathematical equation: $n\in\Z$$ . Then ${(- 1)}^{n} ↓^{n} ○ \partial ○ ↑^{n}$ $Mathematical equation: $(-1)^n\downarrow^n\circ \partial \circ \uparrow^n$$ is a differential on V [n] and $\gl (V) \to \gl (V [n]), g \mapsto {(- 1)}^{n | g |} ↓^{n} ○ g ○ ↑^{n}$ $Mathematical equation: \[\gl(V)\to\gl(V[n]),\quad g\mapsto (-1)^{n|g|}\downarrow^n\circ g\circ \uparrow^n\]$ is a DGLA homomorphism. The induced representation of L on V [n] is given by $\begin{array}{l} S (L [1]) \otimes V [n] \to V [n], \\ x \otimes ↓^{n} v \mapsto {(- 1)}^{n + n | x |} ↓^{n} ρ (x, v) . \end{array}$ $Mathematical equation: \[ \begin{array}{lll} &&\Sy(L[1])\otimes V[n]\to V[n], \\ && x\otimes {\downarrow^n}v\mapsto (-1)^{n+n|x|}{\downarrow^n}\rho(x,v). \end{array} \]$

4.2 Representations as coderivations

Observe that the map $Δ_{V} ≔ Δ_{S} \otimes \id_{V} : S (L [1]) \otimes V \to S (L [1]) \otimes (S (L [1]) \otimes V)$ $Mathematical equation: $\Delta_V\vcentcolon= \Delta_S\otimes \id_V\colon \Sy(L[1])\otimes V \to \Sy(L[1])\otimes (\Sy(L[1])\otimes V)$$ satisfies $(Δ_{S} \otimes \id_{S \otimes V}) Δ_{V} = ({id}_{S} \otimes Δ_{V}) Δ_{V},$ $Mathematical equation: \begin{equation*}(\Delta_S\otimes \id_{S\otimes V})\Delta_V= (\id_S\otimes \Delta_V)\Delta_V, \end{equation*}$ (36)

which makes $S (L [1]) \otimes V$ $Mathematical equation: $\Sy(L[1])\otimes V$$ into a left $S (L [1])$ $Mathematical equation: $\Sy(L[1])$$ -comodule.

Definition 49. Let $d^{'} \in \Coder (S (L [1]))$ $Mathematical equation: $d'\in \Coder(\Sy(L[1]))$$ be of degree p. A coderivation of $S (L [1]) \otimes V$ $Mathematical equation: $\Sy(L[1])\otimes V$$ extending d′ is a linear map $D : S (L [1]) \otimes V \to S (L [1]) \otimes V$ $Mathematical equation: $D\colon \Sy(L[1])\otimes V \to \Sy(L[1])\otimes V$$ of degree p such that $Δ_{V} ○ D = (d^{'} \otimes \id_{S \otimes V} + \id_{S} \otimes D) Δ_{V} .$ $Mathematical equation: \begin{equation*}\Delta_V\circ D= (d' \otimes \id_{S\otimes V} + \id_S\otimes D) \Delta_V. \end{equation*}$ (37)

Proposition 50 ([6], Proposition 1.5.3, p. 31). Let $d^{'} \in \Coder (S (L [1]))$ $Mathematical equation: $d'\in\Coder(\Sy(L[1]))$$ be of degree p. There is a one-to-one correspondence between coderivations D of $S (L [1]) \otimes V$ $Mathematical equation: $\Sy(L[1])\otimes V$$ extending d′ and linear maps $ρ : S (L [1]) \otimes V \to V$ $Mathematical equation: $\rho\colon\Sy(L[1])\otimes V\to V$$ of degree p given by $\begin{array}{l} D & = & d^{'} \otimes \id_{V} + ({id}_{S} \otimes ρ) Δ_{V}, \\ ρ & = & \pr_{V} ○ D, \end{array}$ $Mathematical equation: \begin{eqnarray*}D&=&d'\otimes \id_V + (\id_S\otimes \rho)\Delta_V,\nonumber\\ \rho&=&\pr_V\circ D,\nonumber \end{eqnarray*}$ where $\pr_{V} : S (L [1]) \otimes V \to V$ $Mathematical equation: $\pr_V\colon \Sy(L[1])\otimes V\to V$$ is the projection of $S (L [1]) \otimes V$ $Mathematical equation: $\Sy(L[1]) \otimes V$$ onto $k \otimes V ≅ V$ $Mathematical equation: ${\mathbb k} \otimes V \cong V$$ .

Proof. Let D be a coderivationof $S (L [1]) \otimes V$ $Mathematical equation: $\Sy(L[1])\otimes V$$ extending d′. As (id_S ⊗pr_V)(Δ_S ⊗id_V) = id_S ⊗id_V, we obtain from (37) that $\begin{array}{l} D & = & ({id}_{S} \otimes \pr_{V}) Δ_{V} ○ D \\ = & ({id}_{S} \otimes \pr_{V}) (d^{'} \otimes \id_{S} \otimes \id_{V} + \id_{S} \otimes D) Δ_{V} \\ = & (d^{'} \otimes \id_{V}) ({id}_{S} \otimes \pr_{V}) Δ_{V} + ({id}_{S} \otimes (\pr_{V} ○ D)) Δ_{V} \\ = & d^{'} \otimes \id_{V} + ({id}_{S} \otimes (\pr_{V} ○ D)) Δ_{V} . \end{array}$ $Mathematical equation: \begin{eqnarray*} D&=& (\id_S\otimes \pr_V) \Delta_V \circ D\nonumber\\ &=& (\id_S\otimes \pr_V) (d'\otimes \id_S\otimes \id_V + \id_S\otimes D) \Delta_V\nonumber\\ &=&(d'\otimes \id_V)(\id_S\otimes \pr_V)\Delta_V + (\id_S\otimes (\pr_V\circ D)) \Delta_V\nonumber\\ &=&d'\otimes \id_V + (\id_S\otimes (\pr_V\circ D)) \Delta_V.\nonumber \end{eqnarray*}$

This shows that D is completely determined by pr_V ○ D.

Let conversely $ρ \in \Hom (S (L [1]) \otimes V, V)$ $Mathematical equation: $\rho\in \Hom(\Sy(L[1])\otimes V,V)$$ be of degree p. Using (36) and that d′ is a coderivation, we compute $\begin{array}{l} Δ_{V} ○ ({id}_{S} \otimes ρ) Δ_{V} & = & (Δ_{S} \otimes ρ) Δ_{V} \\ = & ({id}_{S \otimes S} \otimes ρ) ({id}_{S} \otimes Δ_{V}) Δ_{V} \\ = & ({id}_{S} \otimes ({id}_{S} \otimes ρ) Δ_{V}) Δ_{V}, \\ Δ_{V} ○ (d^{'} \otimes \id_{V}) & = & ((d^{'} \otimes \id_{S} + {id}_{S} \otimes d^{'}) Δ_{S}) \otimes \id_{V} \\ = & (d^{'} \otimes \id_{S \otimes V} + \id_{S} \otimes d^{'} \otimes \id_{V}) Δ_{V}, \end{array}$ $Mathematical equation: \begin{eqnarray*} \Delta_V\circ (\id_S\otimes \rho)\Delta_V&=&(\Delta_S\otimes \rho)\Delta_V\nonumber\\ &=&(\id_{S\otimes S}\otimes \rho)(\id_S\otimes \Delta_V)\Delta_V\nonumber\\ &=&(\id_S\otimes (\id_S\otimes \rho)\Delta_V)\Delta_V,\nonumber\\ \Delta_V\circ (d'\otimes \id_V)&=& ((d'\otimes \id_S+\id_S\otimes d')\Delta_S)\otimes \id_V\nonumber\\ &=& (d'\otimes \id_{S\otimes V}+ \id_S \otimes d' \otimes \id_V)\Delta_V,\nonumber \end{eqnarray*}$

which, combined, show that D := d ⊗id_V + (id_S ⊗ ρ)Δ_V is a coderivation of $S (L [1]) \otimes V$ $Mathematical equation: $\Sy(L[1])\otimes V$$ extending d. It is easy to see that then pr_V ○ D = ρ, which completes the proof.

Corollary 51. There is a one-to-one correspondence between representations (up to homotopy) of L on V and coderivations $D : S (L [1]) \otimes V \to S (L [1]) \otimes V$ $Mathematical equation: $D\colon \Sy(L[1])\otimes V\to \Sy(L[1])\otimes V$$ extending d such that D² = 0.

Proof. It is a straightforward computation to check that $D^{2} = \frac{1}{2} [D, D]$ $Mathematical equation: $D^2=\frac{1}{2}[D,D]$$ is a coderivation of $S (L [1]) \otimes V$ $Mathematical equation: $\Sy(L[1])\otimes V$$ extending $\frac{1}{2} [d, d] = d^{2} = 0$ $Mathematical equation: $\frac{1}{2}[d,d]=d^2=0$$ . By Proposition 50, D² = 0 if and only if ρ = pr_V ○ D satisfies $0 = \pr_{V} ○ D^{2} = ρ (d \otimes \id_{V}) + ρ ({id}_{S} \otimes ρ) (Δ_{S} \otimes \id_{V}) .$ $Mathematical equation: \[ 0=\pr_V\circ D^2=\rho(d\otimes \id_V) + \rho(\id_S\otimes \rho)(\Delta_S\otimes \id_V). \]$

4.3 A first approach to L_∞-algebra cohomology

Assume now that the L_∞-algebra L is $ℤ_{\leq 0}$ $Mathematical equation: $\Z_{\leq 0}$$ -graded and of finite type and that V is either finite-dimensional or of finite type and trivial in the negative degrees. We then have $S {(L [1])}^{*} ≅ S (L {[1]}^{*})$ $Mathematical equation: $\Sy(L[1])^*\cong \Sy(L[1]^*)$$ , V ≅ V ^** and ${(S (L [1]) \otimes V^{*})}^{*} ≅ S (L {[1]}^{*}) \otimes V$ $Mathematical equation: $(\Sy(L[1])\otimes V^*)^*\cong\Sy(L[1]^*)\otimes V$$ . Let d_CE = −d^* denote the differential on $S (L {[1]}^{*})$ $Mathematical equation: $\Sy(L[1]^*)$$ . The map $\begin{array}{l} S (L {[1]}^{*}) \otimes (S (L {[1]}^{*}) \otimes V) \to S (L {[1]}^{*}) \otimes V, \\ (ξ \otimes (η \otimes v)) \mapsto (ξ \lor η) \otimes v \end{array}$ $Mathematical equation: \[ \begin{array}{lll} &&\Sy(L[1]^*)\otimes (\Sy(L[1]^*)\otimes V)\to \Sy(L[1]^*)\otimes V,\\ &&\quad(\xi\otimes(\eta\otimes v))\mapsto (\xi\vee \eta)\otimes v \end{array} \]$

makes $S (L {[1]}^{*}) \otimes V$ $Mathematical equation: $\Sy(L[1]^*)\otimes V$$ into a left $S (L {[1]}^{*})$ $Mathematical equation: $\Sy(L[1]^*)$$ -module. Similarly to Definition 49, we call a linear map $D_{C E} : S (L {[1]}^{*}) \otimes V \to S (L {[1]}^{*}) \otimes V$ $Mathematical equation: $D_{CE}\colon \Sy(L[1]^*)\otimes V\to \Sy(L[1]^*)\otimes V$$ of degree one a derivation of $S (L {[1]}^{*}) \otimes V$ $Mathematical equation: $\Sy(L[1]^*)\otimes V$$ extending d_CE if $D_{C E} (ξ \lor (η \otimes v)) = d_{C E} ξ \lor (η \otimes v) + {(- 1)}^{| ξ |} ξ \lor D_{C E} (η \otimes v)$ $Mathematical equation: \[ D_{CE}(\xi\vee (\eta\otimes v))=d_{CE}\xi\vee (\eta\otimes v) +(-1)^{|\xi|}\xi\vee D_{CE}(\eta\otimes v) \]$

holds for all $ξ, η \in S (L {[1]}^{*})$ $Mathematical equation: $\xi,\eta\in \Sy(L[1]^*)$$ , v ∈ V homogeneous.

Note that a representation of L on V is equivalent to a representation on V ^* by Remark 48 and V ≅V ^**. As the notion of a derivation extending d_CE is dual to the one of a coderivation extending d, we get the following dualized version of Corollary 51.

Proposition 52. A representation ρ of L on V is equivalent to a derivation $D_{C E} : S (L {[1]}^{*}) \otimes V \to S (L {[1]}^{*}) \otimes V$ $Mathematical equation: $D_{CE}\colon \Sy(L[1]^*)\otimes V\to \Sy(L[1]^*)\otimes V$$ extending d_CE with $D_{C E}^{2} = 0$ $Mathematical equation: $D_{CE}^2=0$$ . Explicitly, we have D_CE = −D^*, where D is the coderivation extending d induced by the dual representation ρ^∨.

For a fixed representation ρ of L on V , we can then see $S (L {[1]}^{*}) \otimes V$ $Mathematical equation: $\Sy(L[1]^*)\otimes V$$ as our generalized Chevalley–Eilenberg complex with coboundary operator D_CE.

4.4 A dead-end

This not only provides us with an explicit construction of the coboundary operator from a given representation, but also gives it the additional structure of a derivation extending d_CE . Unfortunately, this came at the cost of the finiteness assumptions we imposed on L on V at the beginning of Section 4.3. As our goal is to establish a generalisation of Theorem 32 – which does not need such assumptions – in terms of L_∞-algebra cohomology, this is not the appropriate framework for our purposes. We can, however, make the following observation.

Remark 53. With our finiteness assumptions on L and V , we have $S (L {[1]}^{*}) \otimes V ≅ Hom (S (L [1]), V),$ $Mathematical equation: $ \Sy(L[1]^*)\otimes V\cong\Hom(\Sy(L[1]),V), $$ where $ξ \otimes v \in S (L {[1]}^{*}) \otimes V$ $Mathematical equation: $\xi\otimes v\in \Sy(L[1]^*)\otimes V$$ is identified with the linear map $S (L [1]) \to V$ $Mathematical equation: $\Sy(L[1])\to V$$ , $x \mapsto {(- 1)}^{| x | | v |} ξ (x) \cdot v$ $Mathematical equation: $x\mapsto (-1)^{|x||v|}\xi(x) \cdot v$$ . For $f \in \Hom (S (L [1]), V)$ $Mathematical equation: $f\in \Hom(\Sy(L[1]),V)$$ homogeneous, one finds that D_CE f is then given by $D_{C E} f = ρ ({id}_{S} \otimes f) Δ_{S} - {(- 1)}^{| f |} f ○ d .$ $Mathematical equation: \begin{equation*}D_{CE}f= \rho(\id_S\otimes f)\Delta_S-(-1)^{|f|}f\circ d. \end{equation*}$ (38)

One could then simply define $D_{C E} : \Hom (S (L [1]), V) \to \Hom (S (L [1]), V)$ $Mathematical equation: $D_{CE}\colon \Hom(\Sy(L[1]),V)\to \Hom(\Sy(L[1]),V)$$ by (38), even if L and V do not meet our finiteness assumptions. Although there is a priori no reason for $D_{C E}^{2} = 0$ $Mathematical equation: $D_{CE}^2=0$$ to hold in the general case, a straightforward computation shows that it actually does. While this leaves us with nothing but the formula (38) to work with, it also suggests that there should be another approach to L_∞ -algebra cohomolgy that gets by without the need of finiteness assumptions.

In [4], the L_∞-algebra cohomology with values in the adjoint representation was introduced in terms of the commutator bracket of coderivations and the isomorphism $\Coder (\rsym (L [1]), \rsym (L [1])) ≅ \Hom (\rsym (L [1]), L [1])$ $Mathematical equation: $\Coder(\rsym(L[1]),\rsym(L[1]))\cong \Hom(\rsym(L[1]),L[1])$$ . In the next section, we extend this approach to arbitrary representations, which leads to a generalisation of Theorem 32 in a rather natural way.

5 L_∞-algebra cohomology

5.1 The Lie bracket on $Hom (\rsym (L [1] \oplus V), L [1] \oplus V)$ $Mathematical equation: $\Hom(\rsym(L[1]\oplus V),L[1]\oplus V)$$

Recall from Proposition 21 that $\Coder (S (L [1]))$ $Mathematical equation: $\Coder(\Sy(L[1]))$$ is closed under the graded commutator. Together with Theorem 22, this induces a Lie bracket on $Hom (S (L [1]), L [1])$ $Mathematical equation: $\Hom(\Sy(L[1] ),L[1] )$$ . Its explicit formula is $[f, g] = f ○ μ_{S} (g \otimes {id}_{S}) Δ_{S} - {(- 1)}^{| f | | g |} g ○ μ_{S} (f \otimes \id_{S}) Δ_{S}$ $Mathematical equation: \begin{equation*}[f,g]=f\circ \mu_S(g\otimes\id_S)\Delta_S-(-1)^{|f||g|}g\circ \mu_S(f\otimes \id_S)\Delta_S \end{equation*}$ (39)

for $f, g \in Hom (S (L [1]), L [1])$ $Mathematical equation: $f,g\in\Hom(\Sy(L[1] ),L[1] )$$ homogeneous.

As L_∞-structures correspond to codifferentials with d(1) = 0 and elements in $Hom (\rsym (L [1]), L [1])$ $Mathematical equation: $\Hom(\rsym(L[1]),L[1])$$ , it is only natural to restrict ourselves to the Lie subalgebra $Hom (\rsym (L [1]), L [1])$ $Mathematical equation: $\Hom(\rsym(L[1]),L[1])$$ . Keeping the $Hom (k, L [1])$ $Mathematical equation: $\Hom({\mathbb k}, L[1])$$ part corresponds to the framework of curved L∞-algebras, which are L_∞-algebras that also allow for a 0-ary bracket $k \to L [1]$ $Mathematical equation: ${\mathbb k}\to L[1]$$ .

Remark 54. The same construction also makes $Hom (\rsym (L [1] \oplus V), L [1] \oplus V)$ $Mathematical equation: $\Hom(\rsym(L[1]\oplus V),L[1]\oplus V)$$ into a graded Lie algebra. The decomposition $S (L [1] \oplus V) ≅ S (L [1]) \otimes S (V)$ $Mathematical equation: $\Sy(L[1]\oplus V)\cong \Sy (L[1]) \otimes \Sy (V)$$ implies that $\rsym (L [1] \oplus V) ≅ \rsym (L [1]) \otimes \rsym (V) \oplus \rsym (L [1]) \oplus \rsym (V) .$ $Mathematical equation: \begin{equation*}\rsym(L[1]\oplus V)\cong \rsym(L[1])\otimes \rsym(V)\oplus \rsym(L[1])\oplus \rsym(V). \end{equation*}$ (40)

We can then consider spaces like $Hom (\rsym (L [1]), L [1])$ $Mathematical equation: $\Hom(\rsym(L[1]),L[1])$$ and $Hom (\rsym (L [1]) \otimes V, V)$ $Mathematical equation: $\Hom(\rsym (L[1])\otimes V, V)$$ as subspaces of $Hom (\rsym (L [1] \oplus V), L [1] \oplus V)$ $Mathematical equation: $\Hom(\rsym (L[1] \oplus V),\ L[1]\oplus V)$$ in the obvious way. The inclusion of $Hom (\rsym (L [1]), L [1])$ $Mathematical equation: $\Hom(\rsym(L[1]),L[1])$$ into $Hom (\rsym (L [1] \oplus V), L [1] \oplus V)$ $Mathematical equation: $\Hom(\rsym(L[1]\oplus V),L[1]\oplus V)$$ is then easily seen to preserve the Lie bracket.

Remark 55. In terms of the Lie bracket on $Hom (\rsym (L [1]), L [1])$ $Mathematical equation: $\Hom(\rsym(L[1]),L[1])$$ , the condition (30) for a linear map $λ : \rsym (L [1]) \to L [1]$ $Mathematical equation: $\lambda\colon \rsym(L[1])\to L[1]$$ of degree one to define an L_∞-algebra structure on L[1] becomes $\frac{1}{2} [λ, λ] = 0.$ $Mathematical equation: \begin{equation*} \frac{1}{2}[\lambda,\lambda]=0. \end{equation*}$ (41)

By Example 12 and Remark 54, this makes $Hom (\rsym (L [1] \oplus V), L [1] \oplus V)$ $Mathematical equation: $\Hom(\rsym(L[1]\oplus V),L[1]\oplus V)$$ into a DGLA. Solutions of the Maurer–Cartan equation then induce new L_∞ -structures on L[1] ⊕ V by Example 14.

By abuse of notation, we now denote the (co)products on $S (L [1])$ $Mathematical equation: $\Sy(L[1])$$ and $S (L [1] \oplus V)$ $Mathematical equation: $\Sy(L[1]\oplus V)$$ both by μ_S and Δ_S . This is justified, as they coincide on $S (L [1]) \subset S (L [1] \oplus V)$ $Mathematical equation: $\Sy(L[1])\subset \Sy(L[1]\oplus V)$$ .

In (38), d = μ_S(λ ⊗id_S)Δ_S and μ_S(id_S ⊗ f)Δ_S = μ_S(f ⊗id_S)Δ_S due to $S (L [1])$ $Mathematical equation: $\Sy(L[1])$$ being (co)commutative. The similarity between (38) and (39) suggests to approach L_∞ -algebra cohomology using the Lie bracket on $Hom (\rsym (L [1] \oplus V), L [1] \oplus V)$ $Mathematical equation: $\Hom(\rsym(L[1]\oplus V),L[1]\oplus V)$$ .

Proposition 56. Let $ρ \in Hom (S (L [1]) \otimes V, V)$ $Mathematical equation: $\rho\in\Hom(\Sy(L[1])\otimes V,V)$$ be of degree one. Then ρ is a representation of L on V if and only if $ρ ○ μ_{S} (ρ \otimes \id_{S}) Δ_{S} + ρ ○ μ_{S} (λ \otimes \id_{S}) Δ_{S} = 0,$ $Mathematical equation: \begin{equation*}\rho\circ\mu_S(\rho\otimes \id_S)\Delta_S + \rho\circ\mu_S(\lambda\otimes \id_S)\Delta_S=0, \end{equation*}$ (42) where λ and ρ are considered as elements of $Hom (\rsym (L [1] \oplus V), L [1] \oplus V)$ $Mathematical equation: $\Hom(\rsym(L[1]\oplus V),L[1]\oplus V)$$ .

Proof. Note that ρ ○ μ_S(ρ ⊗id_S)Δ_S and ρ ○ μ_S(ρ ⊗id_S)Δ_S are only possibly nonzero on $S (L [1]) \otimes V$ $Mathematical equation: $\Sy(L[1])\otimes V$$ . For x₁, …, x_n−1 ∈ L[1] and x_n ∈ V , a routine computation using Lemma 24 shows that $\begin{array}{l} (ρ ○ μ_{S} (λ \otimes \id_{S}) Δ_{S}) (x_{1} \lor … \lor x_{n}) \\ = ρ (d (x_{1} \lor … \lor x_{n - 1}), x_{n}), \\ (ρ ○ μ_{S} ({id}_{S} \otimes ρ) Δ_{S}) (x_{1} \lor … \lor x_{n}) \\ = ρ ({id}_{S} \otimes ρ) (Δ_{S} (x_{1} \lor … \lor x_{n - 1}), x_{n}) . \end{array}$ $Mathematical equation: \[\begin{array}{ll} &(\rho\circ\mu_S(\lambda\otimes \id_S)\Delta_S)(x_1\vee\dotsc \vee x_n)\\ &\quad= \rho(d(x_1\vee\dotsc\vee x_{n-1}),x_n),\\ &(\rho\circ\mu_S(\id_S\otimes \rho)\Delta_S)(x_1\vee\dotsc \vee x_n)\\ &\quad=\rho(\id_S\otimes \rho)(\Delta_S(x_1\vee\dotsc\vee x_{n-1}),x_n). \end{array} \]$

As again μ_S(ρ ⊗id_S)Δ_S = μ_S(id_S ⊗ ρ)Δ_S by (co)commutativity of $\rsym (L [1] \oplus V)$ $Mathematical equation: $\rsym(L[1]\oplus V)$$ , ρ satisfies (33) if and only if it satisfies (42).

Corollary 57. An element $ρ \in Hom (S (L [1]) \otimes V, V)$ $Mathematical equation: $\rho\in\Hom(\Sy(L[1])\otimes V,V)$$ of degree one is representation of L on V if and only if (L[1] ⊕ V, λ + ρ) is an L_∞-algebra.

Proof. We have $\begin{array}{l} \frac{1}{2} [λ + ρ, λ + ρ] & = & \frac{1}{2} [λ, λ] + [λ, ρ] + \frac{1}{2} [ρ, ρ] \\ = & ρ ○ μ_{S} (λ \otimes \id_{S}) Δ_{S} + ρ ○ μ_{S} (ρ \otimes \id_{S}) Δ_{S} . \end{array}$ $Mathematical equation: \begin{eqnarray*} \frac{1}{2}[\lambda+\rho,\lambda+\rho]&{=}&\frac{1}{2}[\lambda,\lambda]+[\lambda,\rho]+\frac{1}{2}[\rho,\rho]\nonumber\\ &{=}&\rho\circ\mu_S(\lambda\otimes \id_S)\Delta_S{+} \rho\circ\mu_S(\rho\otimes \id_S)\Delta_S.\nonumber \end{eqnarray*}$

Corollary 58. The subspace $Hom (S (L [1]) \otimes V, V)$ $Mathematical equation: $\Hom(\Sy(L[1])\otimes V,V)$$ is invariant under the Lie bracket [⋅, ⋅] and the differential [λ, ⋅ ]. Representations (up to homotopy) of L on V are then exactly the Maurer–Cartan elements in $Hom (S (L [1]) \otimes V, V)$ $Mathematical equation: $\Hom(\Sy(L[1])\otimes V,V)$$ .

By applying Proposition 33 to Corollary 57 and using that a representation on V is equivalent to one on V [1] by Remark 48, we obtain the following.

Proposition 59. A representation of L on V is equivalent to a system of linear maps ρ_k: ∧^k−1L ⊗ V → V of degree 2 − k for k ≥ 1 such that {l_k + ρ_k: ∧^k(L ⊕ V ) → L ⊕ V ∣1 ≤ k < ∞} is an L_∞-structure on L ⊕ V .

Remark 60. It is easy to see that the generalized Jacobi identity (27) for {l_k + ρ_k∣1 ≤ k < ∞} has only to be checked on ∧ Lⁿ⁻¹ ⊗ V for each n ≥ 1. Representations of L_∞-algebras are often defined in terms of these equations, see for example ([8], Definition 5.1) and ([3], Definition 18). Similarly, equation (42) on $S (L [1]) \otimes V$ $Mathematical equation: $\Sy(L[1])\otimes V$$ is easily seen to be the condition imposed on ρ in ([3], Definition 19).

For a fixed representation ρ of L on V , [λ +ρ, ⋅ ] makes $Hom (\rsym (L [1] \oplus V), L [1] \oplus V)$ $Mathematical equation: $\Hom(\rsym(L[1]\oplus V),L[1]\oplus V)$$ into a DGLA. The space $Hom (\rsym (L [1]), V)$ $Mathematical equation: $\Hom(\rsym(L[1]),V)$$ is then an abelian Lie subalgebra that is invariant under [λ + ρ, ⋅ ]. Explicitly, we have for $f \in \Hom (\rsym (L [1]), V)$ $Mathematical equation: $f\in \Hom(\rsym(L[1]),V)$$ homogeneous $[λ + ρ, f] = ρ ({id}_{S} \otimes f) Δ_{S} - {(- 1)}^{| f |} f ○ d .$ $Mathematical equation: \begin{equation*}[\lambda+\rho, f]=\rho(\id_S\otimes f)\Delta_S-(-1)^{|f|}f\circ d. \end{equation*}$ (43)

Definition 61. The map $δ ≔ [λ + ρ, \ea] : \Hom (\rsym (L [1]), V) \to \Hom (\rsym (L [1]), V)$ $Mathematical equation: $\delta\vcentcolon=[\lambda+\rho,\ea]\colon \Hom (\rsym(L[1]),V)\to \Hom(\rsym(L[1]),V)$$ is called the L∞-coboundary operator. The cohomology of the cochain complex $(Hom (\rsym (L [1]), V), δ)$ $Mathematical equation: $(\Hom(\rsym(L[1]),V),\delta)$$ is called the L∞-algebra cohomology with values in V.

Remark 62. For L and V as in Section 4.3, we clearly have δ = D_CE. If L = 𝔤 and V are concentrated in degree zero, the décalage isomorphism (5) implies that $\begin{array}{l} {nn\Hom}_{p} (\rsym (g [1]), V) & ≅ & \prod_{n \geq 1} {Hom}_{p - n} (\Asy^{n} g, V) \\ ≅ & \Hom (\Asy^{p} g, V) \end{array}$ $Mathematical equation: \begin{eqnarray*} \Hom_p(\rsym(\mathfrak g[1]),V)&\cong& \prod_{n\geq 1}\Hom_{p-n}(\Asy^n \mathfrak g,V)\nonumber\\ &\cong& \Hom(\Asy^p\mathfrak g,V)\nonumber \end{eqnarray*}$

for all p ≥ 1. This way, we recover the usual Lie algebra cohomology.

Example 63 (The adjoint representation). The adjoint representation of L on L[1] is given by $S (L [1]) \otimes L [1] \to L [1]$ $Mathematical equation: $\Sy(L[1])\otimes L[1]\to L[1]$$ , x ⊗ y↦λ(x ∨ y). While there are now two distinct copies of L[1] involved, it is evident by (43) that δ = [λ, ⋅ ], the bracket being the one on $Hom (\rsym (L [1]), L [1])$ $Mathematical equation: $\Hom(\rsym(L[1]),L[1])$$ . This is the case discussed in [4].

5.2 L_∞-structures induced by 2-cocycles

The description of L_∞-structures, representations (up to homotopy) and the L_∞-coboundary operator all by the same Lie bracket yields the following generalisation of Theorem 32.

Theorem 64. Let L and V be graded vector spaces and $λ \in Hom (\rsym (L [1]), L [1])$ $Mathematical equation: ${\lambda\in\Hom(\rsym(L[1]),L[1])}$$ , $ρ \in \Hom (S (L [1]) \otimes V, V)$ $Mathematical equation: $\rho\in \Hom(\Sy(L[1])\otimes V,V)$$ and $ω \in \Hom (\rsym (L [1]), V)$ $Mathematical equation: $\omega\in \Hom(\rsym(L[1]),V)$$ be all of degree one. Then (L[1] ⊕ V, λ + ρ + ω) is an L_∞-algebra if and only if (L[1], λ) is an L_∞-algebra, ρ is a representation of L on V and ω is a V-valued cocycle.

Proof. The map $\frac{1}{2} [λ + ρ + ω, λ + ρ + ω] = \frac{1}{2} [λ, λ] + [λ, ρ] + \frac{1}{2} [ρ, ρ] + [λ + ρ, ω]$ $Mathematical equation: $ \frac{1}{2}[\lambda+\rho+\omega,\lambda+\rho+\omega]=\frac{1}{2}[\lambda,\lambda]+[\lambda,\rho]+\frac{1}{2}[\rho,\rho]+[\lambda+\rho,\omega] $$ decomposes itself into linear maps $\begin{array}{l} \frac{1}{2} [λ, λ] & : & S (L [1]) \to L [1], \\ [λ, ρ] + \frac{1}{2} [ρ, ρ] & : & S (L [1]) \otimes V \to V, \\ [λ + ρ, ω] & : & S (L [1]) \to V . \end{array}$ $Mathematical equation: \begin{eqnarray*} \frac{1}{2}[\lambda,\lambda]&\colon& \Sy(L[1])\to L[1],\nonumber\\ {}[\lambda,\rho]+\frac{1}{2}[\rho,\rho]&\colon& \Sy(L[1])\otimes V\to V,\nonumber\\ {}[\lambda+\rho,\omega]&\colon& \Sy(L[1])\to V.\nonumber \end{eqnarray*}$

The assertion then follows from Remark 55, Corollary 58 and the definition of δ.

In terms of antisymmetric brackets, Theorem 64 characterises L_∞-structures on L ⊕ V in which for each $n \in ℕ$ $Mathematical equation: $n\in \N$$ , the n-ary bracket decomposes into linear maps $\begin{array}{l} {nn\Asy}^{n} L & \to & L, \\ {nn\Asy}^{n - 1} L \otimes V & \to & V, \\ \Asy^{n} L & \to & V . \end{array}$ $Mathematical equation: \begin{eqnarray*} \Asy^n L&\to& L,\nonumber\\ \Asy^{n-1}L\otimes V&\to& V,\nonumber\\ \quad \Asy^n L &\to& V.\nonumber \end{eqnarray*}$

These then correspond to cocycles in ${Hom}_{1} (S (L [1]), V [1]) ≅ \Hom_{2} (S (L [1]), V)$ $Mathematical equation: $\Hom_1(\Sy(L[1]),V[1])\cong \Hom_2(\Sy(L[1]),V)$$ . So, it is the 2-cocycles that characterise these L_∞-structures, as in the Lie algebra case (cf. [9], Proposition 7.5.18, p. 202). $\begin{array}{l} [λ_{2}, λ_{m}] (x \lor y) & = & (λ_{2} ○ μ_{S} (λ_{m} \otimes \id_{S}) Δ_{S}) (x \lor y) + {(- 1)}^{| x |} λ_{m} (x \lor d_{2} (y)) \\ = & λ_{2} ○ μ_{S} (δ (x) \otimes \id_{S}) Δ_{S} (y) + {(- 1)}^{| x |} δ (x) (d_{2} (y)) \\ = & [λ_{2}, δ (x)] (y), \\ \frac{1}{2} [λ_{m}, λ_{m}] (x \lor y) & = & \sum {(- 1)}^{| x_{(2)} | | y_{(1)} |} λ_{m} (λ_{m} (x_{(1)} \lor y_{(1)}) \lor x_{(2)} \lor y_{(2)}) \\ = & \sum {(- 1)}^{| x_{(2)} | + | x_{(1)} | | x_{(2)} |} δ (x_{(2)}) (δ (x_{(1)} \lor y_{(1)}) \lor y_{(2)}) \\ = & \frac{1}{2} \sum {(- 1)}^{x_{(1)}} δ (x_{(1)}) (δ (x_{(2)} \lor y_{(1)}) \lor y_{(2)}) \\ + {(- 1)}^{| x_{(2)} | + | x_{(1)} | | x_{(2)} |} δ (x_{(2)}) (δ (x_{(1)} \lor y_{(1)}) \lor y_{(2)}) \\ = & (\frac{1}{2} \bracket ○ (δ \otimes δ) Δ_{S}) (x) (y), \end{array}$ $Mathematical equation: \begin{eqnarray*} [\lambda_2, \lambda_m](x\vee y)&=&\big(\lambda_2 \circ \mu_S(\lambda_m\otimes \id_S)\Delta_S\big)(x\vee y) + (-1)^{|x|} \lambda_m (x\vee d_2(y))\nonumber\\ &=&\lambda_2\circ \mu_S(\delta(x)\otimes \id_S)\Delta_S(y) + (-1)^{|x|}\delta(x)(d_2(y))\nonumber\\ &=&[\lambda_2,\delta(x)](y),\nonumber\\ \frac{1}{2}[\lambda_m,\lambda_m](x\vee y)&=& \sum (-1)^{|x_{(2)}||y_{(1)}|} \lambda_m(\lambda_m (x_{(1)}\vee y_{(1)})\vee x_{(2)}\vee y_{(2)})\nonumber\\ &=& \sum (-1)^{|x_{(2)}|+|x_{(1)}||x_{(2)}|}\delta(x_{(2)})(\delta(x_{(1)}\vee y_{(1)})\vee y_{(2)})\nonumber\\ &=&\frac{1}{2}\sum (-1)^{x_{(1)}}\delta(x_{(1)})(\delta(x_{(2)}\vee y_{(1)})\vee y_{(2)})\nonumber\\ && +(-1)^{|x_{(2)}|+|x_{(1)}||x_{(2)}|}\delta(x_{(2)})(\delta(x_{(1)}\vee y_{(1)})\vee y_{(2)})\nonumber\\ &=&\left(\frac{1}{2}\bracket\circ (\delta\otimes \delta) \Delta_S\right)(x)(y),\nonumber \end{eqnarray*}$

5.3 Extensions of L_∞-algebras

We conclude with a brief discussion of extensions of L_∞-algebras. This puts some constructions we discussed in context. The notions are completely analog to the Lie algebra case, see for example ([9], Sections 5.1.3 and 7.5.2).

A graded subspace I ⊂ L of an L_∞-algebra (L[1], λ) is called an ideal if λ(x ∨ y) ∈ I[1] for all x ∈ I[1] and $y \in S (L [1])$ $Mathematical equation: $y\in \Sy(L[1])$$ . Then L∕I carries a canonical L_∞-structure such that the projection L → L∕I is a strict homomorphism of L_∞-algebras. An ideal I ⊂ L is always an L_∞-subalgebra as in particular λ(x) ∈ I[1] for all $x \in \rsym (I [1])$ $Mathematical equation: $x\in\rsym(I[1])$$ .

Definition 65. An extension of an L_∞-algebra (L₁[1], λ₁) by another L_∞-algebra (L₂[1], λ₂) is an exact sequence of L_∞-algebras and strict homomorphisms $0 \to L_{2} \overset{ι}{\to} L \overset{p}{\to} L_{1} \to 0.$ $Mathematical equation: \begin{equation*}0\to L_2\xrightarrow{\iota} L \xrightarrow{p} L_1 \to 0. \end{equation*}$ (44)

Given such an exact sequence (44), the graded subspace L₂ ≅ker(p) ⊂ L is an ideal and p induces a strict isomorphism L∕L₂≅L₁ of L_∞-algebras.

We then always have L≅L₁ ⊕ L₂ (non-canonically) as graded vector spaces, so we are essentially concerned with L_∞-structures on L₁ ⊕ L₂ such that the canonical maps L₂ → L₁ ⊕ L₂ and L₁ ⊕ L₂ → L₁ are strict L_∞-algebra homomorphisms. With the decomposition (40), we can decompose such an L_∞-structure $λ : \rsym ((L_{1} \oplus L_{2}) [1]) \to (L_{1} \oplus L_{2}) [1]$ $Mathematical equation: $\lambda\colon \rsym((L_1\oplus L_2)[1])\to (L_1\oplus L_2)[1]$$ into linear degree one maps $\begin{array}{l} λ_{1} & : & \rsym (L_{1} [1]) \to L_{1} [1], ω : \rsym (L_{1} [1]) \to L_{2} [1], \\ 0 & : & \rsym (L_{2} [1]) \to L_{1} [1], λ_{2} : \rsym (L_{2} [1]) \to L_{2} [1], \\ 0 & : & \rsym (L_{1} [1]) \otimes \rsym (L_{2} [1]) \to L_{1} [1], \\ λ_{m} & : & \rsym (L_{1} [1]) \otimes \rsym (L_{2} [1]) \to L_{2} [1] . \end{array}$ $Mathematical equation: \begin{eqnarray*} \lambda_1&\colon& \rsym(L_1[1])\to L_1[1],\quad\omega\colon \rsym(L_1[1])\to L_2[1],\nonumber\\ 0&\colon&\rsym(L_2[1])\to L_1[1],\quad \lambda_2\colon \rsym(L_2[1])\to L_2[1],\nonumber\\ 0&\colon& \rsym(L_1[1])\otimes \rsym(L_2[1])\to L_1[1],\nonumber\\ \lambda_m&\colon& \rsym(L_1[1])\otimes \rsym(L_2[1])\to L_2[1].\nonumber \end{eqnarray*}$

5.3.1 Abelian and central extensions

An L_∞-algebra L is called abelian if only its 1-ary bracket is nontrivial.An abelian L_∞-algebra is then nothing else than a DG vector space.

An L_∞-algebra extension L₂ → L → L₁ is called abelian if L₂ is abelian. The L_∞-structures constructed in Theorem 64 are examples of abelian extensions of L by V.

Similarly, an extension L₂ → L → L₁ is called central if λ(x ∨ y) = 0 for x ∈ L₂[1], $y \in \rsym (L [1])$ $Mathematical equation: $y\in \rsym(L[1])$$ . It is immediate that this is the case if and only if L₂ is abelian and λ_m = 0. For abelian L₂ , the central extensions L₂ → L₁ ⊕ L₂ → L₁ are by Theorem 64 characterised by 2-cocycles of L₁ with values in the trivial representation of L₁ on L₂.

5.3.2 Semidirect sums

An L_∞-algebra ((L₁ ⊕ L₂)[1], λ) is said to be a semidirect sum of the L_∞-algebras (L₁[1], λ₁) and (L₂[1], λ₂) if the canonical sequence L₂ → L₁ ⊕ L₂ → L₁ is an L_∞-algebra extension and if the canonical map L₁ → L₁ ⊕ L₂ is a strict homomorphism of L_∞-algebras. This is clearly the case if and only if ω = 0 in the decomposition above. A semidirect sum of L₁ and L₂ is therefore characterised by λ_m. Note that L₁ ⊂ L₁ ⊕ L₂ is an ideal if and only if λ_m = 0. In this case, L₁ ⊕ L₂ carries the L_∞-structure λ₁ + λ₂ and is called the direct sum of L₁ and L₂.

For an arbitrary $λ_{m} \in \Hom_{1} (\rsym (L_{1} [1]) \otimes \rsym (L_{2} [1]), L_{2} [1])$ $Mathematical equation: $\lambda_m\in \Hom_1(\rsym(L_1[1])\otimes \rsym(L_2[1]),L_2[1])$$ , the condition for λ₁ + λ₂ + λ_m to define an L_∞-structure on L₁ ⊕ L₂ becomes $[λ_{1} + λ_{2}, λ_{m}] + \frac{1}{2} [λ_{m}, λ_{m}] = 0.$ $Mathematical equation: \begin{equation*}[\lambda_1+\lambda_2,\lambda_m]+\frac{1}{2}[\lambda_m,\lambda_m]=0. \end{equation*}$ (45)

The isomorphism $Hom (\rsym (L_{1} [1]) \otimes \rsym (L_{2} [1]), L_{2} [1]) ≅ \Hom (\rsym (L_{1} [1]), Hom (\rsym (L_{2} [1]), L_{2} [1]))$ $Mathematical equation: $\Hom(\rsym(L_1[1])\otimes \rsym(L_2[1]),L_2[1])\cong \Hom(\rsym(L_1[1]),\Hom(\rsym(L_2[1]),L_2[1]))$$ allows for the following characterisation of semidirect sums.

Theorem 66. Let $λ_{m} \in \Hom (\rsym (L_{1} [1]) \otimes \rsym (L_{2} [1]), L_{2} [1])$ $Mathematical equation: $\lambda_m\in \Hom( \rsym(L_1[1])\otimes \rsym(L_2[1]),L_2[1])$$ be of degree one. Then λ_m satisfies (45) if and only if the corresponding linear degree one map $δ : \rsym (L_{1} [1]) \to \Hom (\rsym (L_{2} [1]), L_{2} [1])$ $Mathematical equation: $\delta\colon \rsym(L_1[1])\to \Hom(\rsym(L_2[1]),L_2[1])$$ is a weak homomorphism of L_∞-algebras in the sense that it satisfies (35).

Proof. Note that $Hom (\rsym (L_{1} [1]) \otimes \rsym (L_{2} [1]), L_{2} [1])$ $Mathematical equation: $\Hom( \rsym(L_1[1])\otimes \rsym(L_2[1]),L_2[1])$$ is closed under [⋅, ⋅] and [λ₁ + λ₂, ⋅ ]. Therefore, (45) has only to be checked on $\rsym (L_{1} [1]) \otimes \rsym (L_{2} [1])$ $Mathematical equation: $\rsym(L_1[1])\otimes \rsym(L_2[1])$$ . Let d₁ and d₂ denote the codifferentials on $S (L_{1} [1])$ $Mathematical equation: $\Sy(L_1[1])$$ and $S (L_{2} [1])$ $Mathematical equation: $\Sy(L_2[1])$$ , respectively. For $x \in \rsym (L_{1} [1])$ $Mathematical equation: $x\in\rsym(L_1[1])$$ and $y \in \rsym (L_{2} [1])$ $Mathematical equation: $y\in\rsym(L_2[1])$$ , we then compute

and [λ₁, λ_m](x ∨ y) = λ_m(d₁(x) ∨ y) = (δ ○ d₁)(x)(y).

Example 67. The L_∞-structure on L ⊕ V induced by arepresentation of L on V is a semidirect sum. For compliance with Theorem 66, note that $\gl (L_{2} [1]) \subset \Hom (\rsym (L_{2} [1]), L_{2} [1])$ $Mathematical equation: $\gl(L_2[1])\subset \Hom(\rsym(L_2[1]),L_2[1])$$ is a Lie subalgebra.

References

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Cite this article as: Ben Reinhold. L_∞-algebras and their cohomology, Emergent Scientist 3, 4 (2019)

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[R1] T. Lada, J. Stasheff, Introduction to SH Lie algebras for physicists, Int. J. Theor. Phys. 32, 1087–1104 (1993) [CrossRef] [Google Scholar]

[R2] J. Stasheff, Differential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras, in: Quantum groups (Leningrad, 1990), Lecture Notes in Mathematics, Springer, Berlin, 1992, Vol. 1510, pp. 120–137 [CrossRef] [Google Scholar]

[R3] M. Dehling, Shifted L_∞ bialgebras, Master’s thesis, Göttingen University, 2011. http://www.uni-math.gwdg.de/mdehling/publ/ma.pdf [Google Scholar]

[R4] P. Michael, L-infinity algebras and their cohomology (Escholarship, University of California, 1995) [Google Scholar]

[R5] J.C. Baez, A.S. Crans, Higher-dimensional algebra. VI. Lie 2-algebras, Theory Appl. Categ. 12, 492–538 (2004) [Google Scholar]

[R6] J.-L. Loday, B. Vallette, Algebraic operads, in: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, 2012, Vol. 346 [CrossRef] [Google Scholar]

[R7] Marco Manetti. Lectures on deformations of complex manifolds (deformations from differential graded viewpoint). Rend. Mat. Appl., 24, 1–183 (2004). [Google Scholar]

[R8] T. Lada, M. Markl, Strongly homotopy Lie algebras, Commun. Algebra 23, 2147–2161 (1995) [CrossRef] [Google Scholar]

[R9] J. Hilgert, K.-H. Neeb, Structure and geometry of Lie groups, Springer Monographs in Mathematics, Springer, New York, 2012 [CrossRef] [Google Scholar]