© B. Reinhold, published by EDP Sciences, 2019

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 $V\cong \underset{p\in \mathbb{Z}}{{\displaystyle \oplus }}{V}_p$ for a family of vector spaces ${\left\{V_p\right\}}_{p\in \mathbb{Z}}$. 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 $\mathbb{k}$ 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 $p\in \mathbb{Z}$ such that f(V _n) ⊂ W_n+p for all $n\in \mathbb{Z}$. 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$\underset{p\in \mathbb{Z}}{{\displaystyle \oplus }}{\text{Hom}}_p(V,W)$. 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 ${(V\oplus W)}_p=V_p\oplus W_p$. The isomorphism $$V\otimes W\cong {\displaystyle \underset{p\in \mathbb{Z}}{\oplus }}{\displaystyle \underset{i+j=p}{\oplus }}\left(V_i\otimes W_j\right)$$

allows us to define a grading on V ⊗ W by $${\left(V\otimes W\right)}_p={\displaystyle \underset{i+j=p}{\oplus }}\left(V_i\otimes W_j\right).$$

This extends to a grading on $V^{\otimes n}\text{≔}{\otimes }_{i=1}^{n}V$ given by $${(V^{\otimes n})}_p={\displaystyle \underset{i_1+\text{…}+i_n=p}{\oplus }}{V}_{i_1}\otimes \text{…}\otimes V_{i_n}.$$

We denote by τ_V,W the linear degree preserving map $${\tau }_{V,W}:V\otimes W\to W\otimes V,v\otimes w\mapsto {(-1)}^{\left|v\right|\left|w\right|}w\otimes v.$$

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 $$(f\otimes g)(v\otimes v^{\prime })={(-1)}^{\left|v\right|\left|g\right|}f(v)\otimes g(v^{\prime })$$(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 $$(f^{\prime }\otimes g^{\prime })○(f\otimes g)={(-1)}^{\left|g^{\prime }\right|\left|f\right|}(f^{\prime }○f)\otimes (g^{\prime }○g),$$(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 ${(-1)}^{\left|p\right|\left|q\right|}$. 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 $$\begin{array}{ll}\widehat{\epsilon }(s_i)(v_1\otimes \text{…}\otimes v_n)\hfill & ={(-1)}^{\left|v_i\right|\left|v_{i+1}\right|}{v}_1\otimes \text{…}\otimes v_{i-1}\hfill \\ \hfill & \otimes v_{i+1}\otimes v_i\otimes v_{i+2}\otimes \text{…}\otimes v_n,\hfill \\ \widehat{\chi }(s_i)(v_1\otimes \text{…}\otimes v_n)\hfill & =-{(-1)}^{\left|v_i\right|\left|v_{i+1}\right|}{v}_1\otimes \text{…}\otimes v_{i-1}\hfill \\ \hfill & \otimes v_{i+1}\otimes v_i\otimes v_{i+2}\otimes \text{…}\otimes v_n.\hfill \end{array}$$

We call $\widehat{\epsilon }$ and $\widehat{\chi }$ the (graded) symmetric and (graded) antisymmetric action of 𝔖_n on V ^⊗n, respectively. Note that $\widehat{\epsilon }(\sigma )$ is degree preserving for all σ ∈ 𝔖_n as $$\widehat{\epsilon }(\sigma )(v_1\otimes \text{…}\otimes v_n)=\pm v_{\sigma (1)}\otimes \text{…}\otimes v_{\sigma (n)}.$$(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 $$\widehat{\chi }(\sigma )(v_1\otimes \text{…}\otimes v_n)=\chi (\sigma ;v_1,\dots ,v_n)v_{\sigma (1)}\otimes \text{…}\otimes v_{\sigma (n)}.$$

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 $$v_1\otimes \text{…}\otimes v_n-\widehat{\epsilon }(\sigma )(v_1\otimes \text{…}\otimes v_n)$$

for σ ∈ 𝔖_n. The space ${\mathcal{S}}^n(V)\text{≔}{V}^{\otimes n}/U_S$ 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 $$v_1\otimes \text{…}\otimes v_n-\widehat{\chi }(\sigma )(v_1\otimes \text{…}\otimes v_n)$$

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 $$f(v_1,\text{…},v_n)=\epsilon (\sigma )f(v_{\sigma (1)},\text{…},v_{(n)})$$

holds. We can write this conveniently as $f○\widehat{\epsilon }(\sigma )=f$. Similarly, f is called (graded) antisymmetric if $f○\widehat{\chi }(\sigma )=f$ for all σ ∈ 𝔖_n.

Proposition 1. Let f : V ^⊗n → W be a symmetric linear map. There is a unique linear map $\phi :{\mathcal{S}}^n(V)\to W$ such that the following diagram commutes: where ${\pi }_S:V^{\otimes n}\to {\mathcal{S}}^n(V)$ 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 $\phi :{\mathcal{S}}^n(V)\to W$ 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, ${\mathcal{S}}^n(V)$ 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 ${\mathcal{S}}^n(V)$, Proposition 1 yields an isomorphism between the subspace of Hom(V ^⊗n, W) consisting of all symmetric maps and $\text{Hom}({\mathcal{S}}^n(V),W)$. 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 ${\mathcal{S}}^n(V)$ 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 $$\begin{array}{l}\phi :{\mathcal{S}}^n(V)\to V^{\otimes n},\hfill \\ x_1\vee \text{…}\vee x_n\mapsto \frac{1}{n!}{\sum }_{\sigma \in {\mathfrak{S}}_n}\epsilon (\sigma )x_{\sigma (1)}\otimes \text{…}\otimes x_{\sigma (n)}\hfill \end{array}$$

is well-defined and satisfies ${\pi }_S○\phi ={\text{id}}_{{\mathcal{S}}^n(V)}$ and $\phi ○{\pi }_S=\frac{1}{n!}{\sum }_{\sigma \in {\mathfrak{S}}_n}\widehat{\epsilon }(\sigma )$. As the latter is a projection of V ^⊗n onto said subspace, the claim follows. A similar statement clearly holds for ∧ⁿV .

For $n\in \mathbb{Z}$, we defProposition2 ine the graded vector space V [n] to be the vector space V with the grading defined by $V{[n]}_p=V_{p+n}$. 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 ${({\downarrow }^{\otimes k})}^{-1}={(-1)}^{\frac{k(k-1)}{2}}{\uparrow }^{\otimes k}$ as a consequence of (2).

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

$$\widehat{\epsilon }(\sigma )○{\downarrow }^{\otimes n}={\downarrow }^{\otimes n}○\widehat{\chi }(\sigma ).$$(4)

There is then a degree preserving isomorphism $${\mathcal{S}}^n(V[1])\cong \left({\text{Asy}}^{n}V\right)[n].$$(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 $$\text{begin{array}{@{}c@{}c@{}l@{}}nn\&}{\pi }_S○\downarrow \text{\&^}\otimes n:V^{\otimes n}\to \mathcal{S}(V[1]),{(-1)}^{\frac{n(n-1)}{2}}\text{\&}{\pi }_A○\uparrow \text{\&^}\otimes n:{(V[1])}^{\otimes n}\to {\text{ Asy}}^{n}V\text{nnend{array}}$$

are then antisymmetric and symmetric, respectively. The induced linear maps between $\mathcal{S}(V[1])$ 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 $\mathcal{S}(V[1])\cong ({\text{Asy}}^{n}V)[n]$.

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

$$\begin{array}{cc}l& =\uparrow ○\lambda ○{\downarrow }^{\otimes n},\\ \lambda & ={(-1)}^{\frac{n(n-1)}{2}}\downarrow ○l○{\uparrow }^{\otimes n}.\end{array}$$

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 $$\cdots \stackrel{d}{\to }{V}_{n-1}\stackrel{d}{\to }{V}_n\stackrel{d}{\to }{V}_{n+1}\stackrel{d}{\to }\cdots$$

and is called the cohomology of (V, d). We then call $H_n(V)=\text{ frac{{}$ 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 $ab={(-1)}^{\left|a\right|\left|b\right|}ba$ 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 $A/I\cong {\displaystyle {\oplus }_{n\in \mathbb{Z}}}{A}_n/I_n$ 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 $$(a\otimes b)(a^{\prime }\otimes b^{\prime })={(-1)}^{\left|b\right|\left|a^{\prime }\right|}a{a}^{\prime }\otimes b{b}^{\prime }.$$(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 $$T{(V)}_p={\displaystyle \underset{i_1+\text{…}+i_n=p}{\oplus }}{V}_{i_1}\otimes \text{…}\otimes V_{i_n}$$ 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 $v_1\otimes v_2-{(-1)}^{\left|v_1\right|\left|v_2\right|}{v}_2\otimes v_1$. We call $\mathcal{S}(V)\text{≔}T(V)/I_S$ 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 $v_1\otimes v_2+{(-1)}^{\left|v_1\right|\left|v_2\right|}{v}_2\otimes v_1$. We denote the multiplication in $\mathcal{S}(V)$ and ∧ V by ∨ and ∧, respectively. Note that $\mathcal{S}(V)$ and ∧ V also admit an exterior grading or grading by weight: as V ^⊗n ∩ I_S = U_S, we have $\mathcal{S}(V)={\displaystyle {\oplus }_{n\ge 0}}{\mathcal{S}}^n(V)$ 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 $\mathcal{S}(V)\to A$. Applying this to the linear map $$V\oplus W\to \mathcal{S}(V)\otimes \mathcal{S}(W),(v,w)\mapsto v\otimes 1+1\otimes w$$ yields a homomorphism of graded unital algebras $\mathcal{S}(V\oplus W)\to \mathcal{S}(V)\otimes \mathcal{S}(W)$ that is easilyseen to be an isomorphism with inverse $\mathcal{S}(V)\otimes \mathcal{S}(W)\to \mathcal{S}(V\oplus W)$, 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 $${\text{nnSy}}^n(V\oplus W)\cong {\displaystyle \underset{p+q=n}{\oplus }}{\mathcal{S}}^p(V)\otimes {\mathcal{S}}^q(W),$$(7) $${\text{nnAsy}}^n(V\oplus W)\cong {\displaystyle \underset{p+q=n}{\oplus }}{\text{Asy}}^{p}V\otimes {\text{ Asy}}^{q}W.$$(8)

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

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 ${\displaystyle \underset{p\in \mathbb{Z}}{\oplus }}{\text{Der}}_p(A)$. 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 $$[x,[y,z]]=[[x,y],z]+{(-1)}^{\left|x\right|\left|y\right|}[y,[x,z]]=0$$(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 $[a,b]=ab-{(-1)}^{\left|a\right|\left|b\right|}ba$ 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 $\frac{1}{2}[x,x]=0$. 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 $\frac{1}{2}[\partial ,\partial ]={\partial }^2=0$.

Definition 13. For a DGLA (L, [⋅, ⋅], d), a Maurer–Cartan element is an element x ∈ L of degree one such that $$d(x)+\frac{1}{2}[x,x]=0.$$(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 $\frac{1}{2}[x+y,x+y]=0$ 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 ${\text{Sh}}_{i,n-i}^{-1}\subset {\mathfrak{S}}_n$. Using the antisymmetry of the Lie bracket, one can rewrite (9) as $${\sum }_{\sigma \in {\text{Sh}}_{2,1}^{-1}}\chi (\sigma )[[x_{\sigma (1)},x_{\sigma (2)}],x_{\sigma (3)}]=0$$(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 $\tau \in {\text{ Sh}}_{i,n-i}^{-1}$ 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 $\epsilon :C\to \mathbb{k}$ 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 $$(f\otimes f){\Delta }_C={\Delta }_D○f.$$(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 $n\in \mathbb{N}$, 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 $${\Delta }^n(x)={\displaystyle \sum x_{(1)}}\otimes \text{…}\otimes x_{(n+1)}.$$

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

Lemma 16. Let (C, Δ_C) be a coassociative coalgebra. Then for all $p,q\in \mathbb{N}$, $$({\Delta }_C^p\otimes {\Delta }_C^q){\Delta }_C={\Delta }_C^{p+q+1}.$$(15)

If (D, Δ_D) is another coassociative coalgebra and if f : C → D is a coalgebra homomorphism, $$f^{\otimes (n+1)}○{\Delta }_C^n={\Delta }_D^n○f$$(16) holds for all $n\in \mathbb{N}$.

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 $u:\mathbb{k}\to C$. The coproduct on $\mathbb{k}$ is given by $1_{\mathbb{k}}\mapsto 1_{\mathbb{k}}\otimes 1_{\mathbb{k}}$ and its counit is the identity on $\mathbb{k}$. Denoting $u(1_{\mathbb{k}})\in C$ as 1, the conditions for u to be a homomorphism of counital coalgebras become Δ(1) = 1 ⊗ 1 and $\epsilon ○u={\text{id}}_{\mathbb{k}}$. 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 $\text{ol}C\text{≔}\ker (\epsilon )$. We claim that $C\cong \text{ ol}C\oplus \mathbb{k}$. Indeed, as $\epsilon ○u={\text{id}}_{\mathbb{k}}$, the short exact sequence of graded vector spaces $$0\to \ker (\epsilon )�C\stackrel{\epsilon }{\to }\mathbb{k}\to 0$$

splits. For $x\in \text{ ol}C$, we then define $\text{ol}\Delta (x)\text{≔}\Delta (x)-x\otimes 1-1\otimes x$. Using (13) and ε(x) = 0, one easily sees that $$\text{ol}\Delta (x)\in \ker (\epsilon \otimes {\text{ id}}_C)\cap \ker ({\text{id}}_C\otimes \epsilon )=\text{ ol}C\otimes \text{ ol}C;$$

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

Conversely, given a coassociative coalgebra $(\text{ol}C,\text{ ol}\Delta )$, we define a coproduct on $C\text{≔}\text{ ol}C\oplus \mathbb{k}$ by Δ(1) = 1 ⊗ 1 and $\Delta (x)=\text{ol}\Delta (x)+x\otimes 1+1\otimes x$ for $x\in \text{ ol}C$. This makes C into a coaugmented coalgebra; the counit and coaugmentation map are given by the projection $C\to \mathbb{k}$ and the inclusion $\mathbb{k}�C$, 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 $$f=\text{ol}f\oplus {\text{ id}}_{\mathbb{k}}:\text{ ol}C\oplus \mathbb{k}\to \text{ ol}D\oplus \mathbb{k}$$

for a unique degree preserving linear map $\text{ol}f:\text{ ol}C\to \text{ ol}D$. It is then easy to see that f is a homomorphism of coaugmented coalgebras if and only if $\text{ol}f$ is a homomorphism of coalgebras. This yields a one-to-one correspondence between coalgebra homomorphisms $\text{ol}C\to \text{ ol}D$ 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 $x\in \text{ ol}C$ there is an $n\in \mathbb{N}$ such that $\text{ol}{\Delta }^n(x)=0$.

2.4.2 Examples of coalgebras

There is a coproduct $\text{ol}{\Delta }_A$ on $\text{rten}(V)\text{≔}{\displaystyle \underset{n\ge 1}{\oplus }}{V}^{\otimes n}$ given by $$\text{ol}{\Delta }_A(v_1\dots v_n)={\sum }_{i=1}^{n-1}(v_1\dots v_i)\otimes (v_{i+1}\text{…}{v}_n),$$

where we now denote the multiplication in T(V ) by concatenation to avoid ambiguities. This makes $\text{rten}(V)$ into a coassociative graded coalgebra. The induced coaugmented coalgebra (T(V ), Δ_A) is called the tensor coalgebra. Inductively, one finds $$\begin{array}{lll}\hfill & \hfill & \text{ol}{\Delta }_A^m(v_1\dots v_n)\hfill \\ \hfill & \hfill & ={\sum }_{1\le i_1<\text{…}<i_m<n}(v_1\dots v_{i_1})\otimes \text{…}\otimes (v_{i_m+1}\text{…}{v}_n),\hfill \end{array}$$(17) which shows that T(V ) is conilpotent.

Proposition 17. Let (C, Δ) be a conilpotent coalgebra and $f:\text{ ol}C\to V$ a linear degree preserving map. There is a unique homomorphism of coaugmented coalgebras $\tilde{f}:C\to T(V)$ such that $f={\text{ pr}}_V○\tilde{f},$ 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 $\tilde{f}:\text{ ol}C\to \text{ rten}(V)$ satifying ${\text{pr}}_V○\tilde{f}=f$. For the uniqueness, assume that there is such $\tilde{f}$. By Lemma 16, we have $${\tilde{f}}^{\otimes (n+1)}○\text{ ol}{\Delta }^n=\text{ol}{\Delta }_A^n○\tilde{f}$$

for all $n\in \mathbb{N}$. Composing both sides with ${\text{pr}}_V^{\otimes (n+1)}$ and noting that ${\text{pr}}_V^{\otimes (n+1)}○\text{ol}{\Delta }_A^n={\text{ pr}}_{V^{\otimes (n+1)}}$ then yields $${\text{pr}}_{V^{\otimes (n+1)}}○\tilde{f}={\text{pr}}_V^{\otimes (n+1)}○{\tilde{f}}^{\otimes (n+1)}○\text{ol}{\Delta }^n=f^{\otimes (n+1)}○\text{ol}{\Delta }^n.$$

This shows that $\tilde{f}$ is completely determined by f and therefore unique. For the existence, consider the linear map ${\sum }_{n=0}^{\infty }\text{ ol}{\Delta }^n:\text{ ol}C\to \text{ rten}(\text{ol}C).$ This is well-defined as C is conilpotent. A straightforward computation using Lemma 16 and ${\text{ol}{\Delta }_A|}_V=0$ shows that $$\left({\sum }_{n=0}^{\infty }\text{ ol}{\Delta }^n\otimes {\sum }_{n=0}^{\infty }\text{ ol}{\Delta }^n\right)\text{ol}\Delta =\text{ol}{\Delta }_A○{\sum }_{n=0}^{\infty }\text{ ol}{\Delta }^n,$$

so that ${\sum }_{n=0}^{\infty }\text{ ol}{\Delta }^n$ is a coalgebra homomorphism. As $\text{rten}(f)={\displaystyle \underset{n\ge 1}{\oplus }}{f}^{\otimes n}:\text{ rten}(\text{ol}C)\to \text{ rten}(V)$ is easily seen to be also a coalgebra homomorphism, $\tilde{f}\text{≔}\text{ rten}(f)○{\sum }_{n=0}^{\infty }\text{ ol}{\Delta }^n:C\to \text{ rten}(V)$ is a homomorphism of coalgebras with ${\text{pr}}_V○\tilde{f}=f$.

Let $\text{rsym}(V)\text{≔}{\displaystyle \underset{n\ge 1}{\oplus }}{\mathcal{S}}^n(V)$. Consider the linear maps $$\begin{array}{l}\pi :\text{ rten}(V)\to \text{ rsym}(V),v_1\otimes \text{…}\otimes v_n\mapsto \frac{1}{n!}{v}_1\vee \text{…}\vee v_n,\hfill \\ N:\text{ rsym}(V)\to \text{ rten}(V),\hfill \\ v_1\vee \text{…}\vee v_n\mapsto {\sum }_{\sigma \in {\mathfrak{S}}_n}\epsilon (\sigma )v_{\sigma (1)}\otimes \text{…}\otimes v_{\sigma (n)}.\hfill \end{array}$$

It is immediate that π○N = id_S, where here and subsequently, we abbreviate $\text{rsym}(V)$ and $\mathcal{S}(V)$ to S in subscripts. Using Lemma 15, we compute $$\begin{array}{ll}\hfill & \text{ol}{\Delta }_A(N(v_1\vee \dots \vee v_n))\hfill \\ \hfill & ={\sum }_{i=1}^{n-1}{\sum }_{\sigma \in {\mathfrak{S}}_n}\epsilon (\sigma )(v_{\sigma (1)}\text{…}{v}_{\sigma (i)})\otimes (v_{\sigma (i+1)}\text{…}{v}_{\sigma (n)})\hfill \\ \hfill & ={\sum }_{i=1}^{n-1}{\sum }_{\tau \in {\text{Sh}}_{i,n-i}^{-1}}{\sum }_{(\alpha ,\beta )\in {\mathfrak{S}}_i\times {\mathfrak{S}}_{n-i}}\epsilon (\tau (\alpha ,\beta ))(v_{(\tau (\alpha ,\beta ))(1)}\text{…}{v}_{(\tau (\alpha ,\beta ))(i)})\otimes (v_{(\tau (\alpha ,\beta ))(i+1)}\text{…}{v}_{(\tau (\alpha ,\beta ))(n)})\hfill \\ \hfill & ={\sum }_{i=1}^{n-1}{\sum }_{\tau \in {\text{Sh}}_{i,n-i}^{-1}}\epsilon (\tau ){\sum }_{(\alpha ,\beta )\in {\mathfrak{S}}_i\times {\mathfrak{S}}_{n-i}}\widehat{\epsilon }(\alpha )(v_{\tau (1)}\text{…}{v}_{\tau (i)})\otimes \widehat{\epsilon }(\beta )(v_{\tau (i+1)}\text{…}{v}_{\tau )(n)})\hfill \\ \hfill & ={\sum }_{i=1}^{n-1}{\sum }_{\tau \in {\text{Sh}}_{i,n-i}^{-1}}\epsilon (\tau )N(v_1\vee \text{…}\vee v_i)\otimes N(v_{i+1}\vee \text{…}\vee v_n).\hfill \end{array}$$

This shows that $\text{ima}(N)\subset \text{ rten}(V)$ is a subcoalgebra and induces a coproduct on $\text{rsym}(V)\cong \text{ ima}(N)$. As a subcoalgebra of a coassociative coalgebra is clearly coassociative itself, $\mathcal{S}(V)$ becomes a coaugmented coalgebra with the coproduct ${\Delta }_S:\mathcal{S}(V)\to \mathcal{S}(V)\otimes \mathcal{S}(V)$ given by $$\begin{array}{lll}{\Delta }_S(v_1\vee \text{…}\vee v_n)\hfill & =\hfill & {\sum }_{i=0}^n{\sum }_{\tau \in {\text{Sh}}_{i,n-i}^{-1}}\epsilon (\tau )(v_{\tau (1)}\vee \text{…}\vee v_{\tau (i)})\hfill \\ \hfill & \hfill & \otimes (v_{\tau (i+1)}\vee \text{…}\vee v_{\tau (n)}).\hfill \end{array}$$(18)

It is immediate that $\mathcal{S}(V)$ is conilpotent, as it is a subcoalgebra of T(V ). We claim that $\mathcal{S}(V)$ 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 ${\text{Sh}}_{i,n-i}^{-1}{\sigma }_i={\text{Sh}}_{n-i,i}^{-1}$ and therefore $$\begin{array}{ll}\hfill & {\tau }_{S,S}○{\Delta }_S(v_1\vee \text{…}\vee v_n)\hfill \\ \hfill & ={\sum }_{i=0}^n{\sum }_{\tau \in {\text{Sh}}_{i,n-i}^{-1}}\epsilon (\tau {\sigma }_i)(v_{(\tau {\sigma }_i)(1)}\vee \text{…}\vee v_{(\tau {\sigma }_i)(n-i)})\hfill \\ \hfill & \otimes (v_{(\tau {\sigma }_i)(n-i+1)}\vee \text{…}\vee v_{(\tau {\sigma }_i)(n)})\hfill \\ \hfill & ={\sum }_{i=0}^n{\sum }_{\tau \in {\text{Sh}}_{n-i,i}^{-1}}\epsilon (\tau )(v_{\tau (1)}\vee \text{…}\vee v_{\tau (i)})\hfill \\ \hfill & \otimes (v_{\tau (i+1)}\vee \text{…}\vee v_{\tau (n)})\hfill \\ \hfill & ={\Delta }_S(v_1\vee \text{…}\vee v_n).\hfill \end{array}$$

Proposition 18. Let (C, Δ) be a cocommutative conilpotent coalgebra and $f:\text{ ol}C\to V$ a degree preserving linear map. There is a unique homomorphism of coaugmented coalgebras $\tilde{f}:C\to \mathcal{S}(V)$ such that $f={\text{ pr}}_V○\tilde{f}.$

Proof. As in the proof of Proposition 17, it suffices to show that there is a unique homomorphism of coalgebras $\tilde{f}:\text{ ol}C\to \text{ rsym}(V)$ satisfying ${\text{pr}}_V○\tilde{f}=f$. Recall that $\text{ol}T(f)○{\sum }_n\text{ ol}{\Delta }^n$ is the unique coalgebra homomorphism $\text{ol}C\to \text{ ol}T(V)$ extending f. For 0 ≤ i ≤ n − 1, we have $$\begin{array}{ll}\hfill & ({\text{id}}_V^{\otimes i}\otimes {\tau }_{C,C}\otimes {\text{ id}}_V^{\otimes (n-i-1)})\text{rten}(f)○\text{ol}{\Delta }^n\hfill \\ \hfill & =\text{rten}(f)({\text{id}}_C^{\otimes i}\otimes {\tau }_{C,C}\otimes {\text{ id}}_C^{\otimes (n-i-1)})\text{ol}{\Delta }^n\hfill \\ \hfill & =\text{ rten}(f)({\text{id}}_C^{\otimes i}\otimes ({\tau }_{C,C}○\text{ol}\Delta )\otimes {\text{ id}}_C^{\otimes (n-i-1)})\text{ol}{\Delta }^{n-1}\hfill \\ \hfill & =\text{ rten}(f)({\text{id}}_C^{\otimes i}\otimes \text{ ol}\Delta \otimes {\text{ id}}_C^{\otimes (n-i-1)})\text{ol}{\Delta }^{n-1}\hfill \\ \hfill & =\text{ rten}(f)○\text{ol}{\Delta }^n\hfill \end{array}$$

since C is cocommutative. As $({\text{id}}_V^{\otimes i}\otimes {\tau }_{C,C}\otimes {\text{ id}}_V^{\otimes (n-i-1)})=\widehat{\epsilon }(s_i),$ the image of $\text{rten}(f)○{\sum }_n\text{ol}{\Delta }^n$ is contained in the subspace of $\text{rten}(V)$ of all symmetric elements, which is im(N). We obtain an induced homomorphism of coalgebras $$\begin{array}{l}\tilde{f}=\pi ○\text{ ol}T(f)○{\sum }_{n=0}^{\infty }\text{ol}{\Delta }^n:\text{ ol}C\to \text{ rsym}(V),\hfill \\ x\mapsto {\sum }_{n=1}^{\infty }\frac{1}{n!}{\displaystyle \sum f}(x_{(1)})\vee \text{…}\vee f(x_{(n)})\hfill \end{array}$$(19)

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

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

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 $$(\Delta \otimes {\text{ id}}_M){\Delta }_l=({\text{id}}_C\otimes {\Delta }_l){\Delta }_l.$$(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 $$({\text{id}}_M\otimes \Delta ){\Delta }_r=({\Delta }_r\otimes {\text{ id}}_C){\Delta }_r.$$(21)

If M is both a left and a right comodule over C and if the compatibility relation $$({\Delta }_l\otimes {\text{ id}}_C){\Delta }_r=({\text{id}}_C\otimes {\Delta }_r){\Delta }_l$$(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 $$\Delta ○d=(d\otimes {\text{ id}}_C){\Delta }_r+({\text{id}}_C\otimes d){\Delta }_l.$$(23)

We denote the vector space of all these maps by Coder_p(M, C) and by Coder(M, C) the graded vector space${\displaystyle \underset{p\in \mathbb{Z}}{\oplus }}{\text{ Coder}}_p(M,C)$.

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 $\text{ol}f:\text{ ol}D\to \text{ ol}C$ then makes $\text{ol}D$ into a comodule over $\text{ol}C$. The following proposition relates elements in $\text{Coder}(\text{ol}D,\text{ol}C)$ 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 $\text{ol}d:\text{ ol}D\to \text{ ol}C$ given by $d=\text{ol}d\oplus 0:\text{ ol}D\oplus \mathbb{k}\to \text{ ol}C\oplus \mathbb{k}$.

Proof. Given a linear map $\text{ol}d\in \text{Hom}(\text{ol}D,\text{ol}C)$, one easily checksthat $\text{ol}d$ is a coderivation if and only if$\text{ol}d\oplus 0:\text{ ol}D\oplus \mathbb{k}\to \text{ ol}C\oplus \mathbb{k}$ 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 ${\mu }_{\mathbb{k}}:\mathbb{k}\otimes \mathbb{k}\to \mathbb{k}$ the multiplication on $\mathbb{k}$. From (13), (23) and the compatibility with the counit (24) it then follows that $$\begin{array}{ll}\epsilon ○d\hfill & ={\mu }_{\mathbb{k}}(\epsilon \otimes \epsilon ){\Delta }_C○d\hfill \\ \hfill & ={\mu }_{\mathbb{k}}(\epsilon \otimes (\epsilon ○d)){\Delta }_l+{\mu }_{\mathbb{k}}((\epsilon ○d)\otimes \epsilon ){\Delta }_r\hfill \\ \hfill & =2(\epsilon ○d),\hfill \end{array}$$

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

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 $f:D\to \mathcal{S}(V)$ a homomorphism of coaugmented coalgebras. The linear map $$\text{Coder}(D,\mathcal{S}(V))\to \text{ Hom}(D,V),d\mapsto {\text{ pr}}_V○d$$ is then an isomorphism. Its inverse is given by $$\text{Hom}(D,V)\to \text{ Coder}(D,\mathcal{S}(V)),\lambda \mapsto {\mu }_S(\lambda \otimes f){\Delta }_D,$$ where ${\mu }_S:\mathcal{S}(V)\otimes \mathcal{S}(V)\to \mathcal{S}(V)$ denotes the multiplication on $\mathcal{S}(V)$ and Δ_D the coproduct on D.

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

The first part of Theorem 22 actually holds for a broader class of comodules over $\mathcal{S}(V)$ (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 $\tau \in {\text{Sh}}_{i,n-i}^{-1}$ either τ(i) = n or τ(n) = n. In the first case, there is a unique $\sigma \in {\text{Sh}}_{i-1,n-i}^{-1}$ 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 $\sigma \in {\text{Sh}}_{i,n-i-1}^{-1}$. This yields abijection ${\text{Sh}}_{i,n-i}^{-1}\cong {\text{Sh}}_{i-1,n-i}^{-1}\bigsqcup {\text{ Sh}}_{i,n-i-1}^{-1}$. By setting ${\text{Sh}}_{-1,n}^{-1},{\text{Sh}}_{n,-1}^{-1}=\varnothing$, this also holds for i = 0, n.

Lemma 24. The map ${\Delta }_S:\mathcal{S}(V)\to \mathcal{S}(V)\otimes \mathcal{S}(V)$ is a homomorphism of graded algebras.

Proof. We show by induction over $n\in \mathbb{N}$ that $${\Delta }_S(v_1\vee \text{…}\vee v_n)={\Delta }_S(v_1)\text{…}{\Delta }_S(v_n).$$(25)

For n = 1 there is nothing to do. Assume that (25) holds for n ≥ 1. We compute $$\begin{array}{ll}\hfill & {\Delta }_S(v_1)\text{…}{\Delta }_S(v_{n+1})\hfill \\ \hfill & ={\Delta }_S(v_1\vee \text{…}\vee v_n){\Delta }_S(v_{n+1})\hfill \\ \hfill & ={\sum }_{i=0}^n{\sum }_{\tau \in {\text{Sh}}_{i,n-i}^{-1}}\left(\epsilon (\tau )(v_{\tau (1)}\vee \text{…}\vee v_{\tau (i)})\otimes (v_{\tau (i+1)}\vee \text{…}\vee v_{\tau (n)})\right)(v_{n+1}\otimes 1+1\otimes v_{n+1})\hfill \\ \hfill & ={\sum }_{i=0}^n{\sum }_{\sigma \in {\text{Sh}}_{i,n-i}^{-1}}{(-1)}^{\left|v_{n+1}\right|{\sum }_{k=i+1}^n\left|v_{\sigma (k)}\right|}\epsilon (\sigma )(v_{\sigma (1)}\vee \text{…}\vee v_{\sigma (i)}\vee v_{n+1})\otimes (v_{\sigma (i+1)}\vee \text{…}\vee v_{\sigma (n)})\hfill \\ \hfill & +{\sum }_{i=0}^n{\sum }_{\sigma \in {\text{Sh}}_{i,n-i}^{-1}}\epsilon (\sigma )(v_{\sigma (1)}\vee \text{…}\vee v_{\sigma (i)})\otimes (v_{\sigma (i+1)}\vee \text{…}\vee v_{\sigma (n)}\vee v_{n+1})\hfill \\ \hfill & ={\sum }_{i=0}^{n+1}{\sum }_{\sigma \in {\text{Sh}}_{i,n+1-i}^{-1}}\epsilon (\sigma )(v_{\sigma (1)}\vee \text{…}\vee v_{\sigma (i)})\otimes (v_{\sigma (i+1)}\vee \text{…}\vee v_{\sigma (n+1)})\hfill \\ \hfill & ={\Delta }_S(v_1\vee \text{…}\vee v_{n+1}).\hfill \end{array}$$