© 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 for a family of vector spaces . 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 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 such that f(V _n) ⊂ W_n+p for all . 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. 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 . The isomorphism

allows us to define a grading on V ⊗ W by

This extends to a grading on given by

We denote by τ_V,W the linear degree preserving map

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 (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 (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 . 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

We call and the (graded) symmetric and (graded) antisymmetric action of 𝔖_n on V ^⊗n, respectively. Note that is degree preserving for all σ ∈ 𝔖_n as (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

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

for σ ∈ 𝔖_n. The space 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

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

holds. We can write this conveniently as . Similarly, f is called (graded) antisymmetric if for all σ ∈ 𝔖_n.

Proposition 1. Let f : V ^⊗n → W be a symmetric linear map. There is a unique linear map such that the following diagram commutes: where 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 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, 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 , Proposition 1 yields an isomorphism between the subspace of Hom(V ^⊗n, W) consisting of all symmetric maps and . 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 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

is well-defined and satisfies and . As the latter is a projection of V ^⊗n onto said subspace, the claim follows. A similar statement clearly holds for ∧ⁿV .

For , we defProposition2 ine the graded vector space V [n] to be the vector space V with the grading defined by . 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 as a consequence of (2).

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

(4)

There is then a degree preserving isomorphism (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

are then antisymmetric and symmetric, respectively. The induced linear maps between 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 .

Corollary 1. There is for each a one-to-one correspondence between symmetric linear maps of degree p and antisymmetric linear maps l: V ^⊗n → V of degree p + 1 − n given by

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

and is called the cohomology of (V, d). We then call 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 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 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 (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 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 . We call 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 . We denote the multiplication in and ∧ V by ∨ and ∧, respectively. Note that and ∧ V also admit an exterior grading or grading by weight: as V ^⊗n ∩ I_S = U_S, we have 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 . Applying this to the linear map yields a homomorphism of graded unital algebras that is easilyseen to be an isomorphism with inverse , 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 (7) (8)

For a graded algebra A, a derivation of A of degree p is a linear map d: A → A of degree p satisfying

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 . 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 (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 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 . 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 .

Definition 13. For a DGLA (L, [⋅, ⋅], d), a Maurer–Cartan element is an element x ∈ L of degree one such that (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 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 . Using the antisymmetry of the Lie bracket, one can rewrite (9) as (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 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 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 (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 , 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

In this notation, for example, the condition for C to be cocommutative becomes for all x ∈ C.

Lemma 16. Let (C, Δ_C) be a coassociative coalgebra. Then for all , (15)

If (D, Δ_D) is another coassociative coalgebra and if f : C → D is a coalgebra homomorphism, (16) holds for all .

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 . The coproduct on is given by and its counit is the identity on . Denoting as 1, the conditions for u to be a homomorphism of counital coalgebras become Δ(1) = 1 ⊗ 1 and . 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 . We claim that . Indeed, as , the short exact sequence of graded vector spaces

splits. For , we then define . Using (13) and ε(x) = 0, one easily sees that

for the last equality, note that We call the reduced coproduct on . It is straightforward to check that is a coassociative coalgebra.

Conversely, given a coassociative coalgebra , we define a coproduct on by Δ(1) = 1 ⊗ 1 and for . This makes C into a coaugmented coalgebra; the counit and coaugmentation map are given by the projection and the inclusion , 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

for a unique degree preserving linear map . It is then easy to see that f is a homomorphism of coaugmented coalgebras if and only if is a homomorphism of coalgebras. This yields a one-to-one correspondence between coalgebra homomorphisms 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 there is an such that .

2.4.2 Examples of coalgebras

There is a coproduct on given by

where we now denote the multiplication in T(V ) by concatenation to avoid ambiguities. This makes into a coassociative graded coalgebra. The induced coaugmented coalgebra (T(V ), Δ_A) is called the tensor coalgebra. Inductively, one finds (17) which shows that T(V ) is conilpotent.

Proposition 17. Let (C, Δ) be a conilpotent coalgebra and a linear degree preserving map. There is a unique homomorphism of coaugmented coalgebras such that 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 satifying . For the uniqueness, assume that there is such . By Lemma 16, we have

for all . Composing both sides with and noting that then yields

This shows that is completely determined by f and therefore unique. For the existence, consider the linear map This is well-defined as C is conilpotent. A straightforward computation using Lemma 16 and shows that

so that is a coalgebra homomorphism. As is easily seen to be also a coalgebra homomorphism, is a homomorphism of coalgebras with .

Let . Consider the linear maps

It is immediate that π○N = id_S, where here and subsequently, we abbreviate and to S in subscripts. Using Lemma 15, we compute

This shows that is a subcoalgebra and induces a coproduct on . As a subcoalgebra of a coassociative coalgebra is clearly coassociative itself, becomes a coaugmented coalgebra with the coproduct given by (18)

It is immediate that is conilpotent, as it is a subcoalgebra of T(V ). We claim that 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 and therefore

Proposition 18. Let (C, Δ) be a cocommutative conilpotent coalgebra and a degree preserving linear map. There is a unique homomorphism of coaugmented coalgebras such that

Proof. As in the proof of Proposition 17, it suffices to show that there is a unique homomorphism of coalgebras satisfying . Recall that is the unique coalgebra homomorphism extending f. For 0 ≤ i ≤ n − 1, we have

since C is cocommutative. As the image of is contained in the subspace of of all symmetric elements, which is im(N). We obtain an induced homomorphism of coalgebras (19)

with . Similarly, a coalgebra homomorphism gives rise to a coalgebra homomorphism that is uniquely determined by by Proposition 17. As N is injective, this shows uniqueness of .

Example 19. A linear degree preserving map f : V → W can be extended by zero to a linear map . The induced homomorphism of coaugmented coalgebras is denoted by and is given by

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 (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 (21)

If M is both a left and a right comodule over C and if the compatibility relation (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 (23)

We denote the vector space of all these maps by Coder_p(M, C) and by Coder(M, C) the graded vector space.

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 then makes into a comodule over . The following proposition relates elements in 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 given by .

Proof. Given a linear map , one easily checksthat is a coderivation if and only if 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 the multiplication on . From (13), (23) and the compatibility with the counit (24) it then follows that

which shows that . Hence, d decomposes as .

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 a homomorphism of coaugmented coalgebras. The linear map is then an isomorphism. Its inverse is given by where denotes the multiplication on and Δ_D the coproduct on D.

It is immediate that d(1) = 0 if and only if λ = pr_V○d vanishes on . Together with Proposition 20, this shows .

The first part of Theorem 22 actually holds for a broader class of comodules over (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 either τ(i) = n or τ(n) = n. In the first case, there is a unique 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 . This yields abijection . By setting , this also holds for i = 0, n.

Lemma 24. The map is a homomorphism of graded algebras.

Proof. We show by induction over that (25)

For n = 1 there is nothing to do. Assume that (25) holds for n ≥ 1. We compute

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

Proof of Theorem 22. Let be a coderivation, that is

Inductively,we then get

For , let be the linear map defined by

From being a subcoalgebra of T(V ) and (17), it follows that . We then have

As this holds for all and as 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

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

2.5 Dual spaces

The graded vector space is called the dual space of V. By degree reasons, . For f ∈ Hom_p(V, W), the linear map f^*∈ Hom_p(W^*, V ^*) is defined by for φ ∈ W^* homogeneous. Note that and if g is a homogeneous linear map with domain W, we have .

We say that V is of finite type if V _k is finite-dimensional for all . 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 -graded. Notions as -graded or -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 the decomposition has only finitely many non-trivial summands, then the canonical inclusion is an isomorphism.

Proof. It is well-known that for finite-dimensional (ungraded) vector spaces V ₁ , …, V _n, the canonical inclusion is an isomorphism. We then have

Remark 26. If V is of finite type and -graded, V ^* is also of finite type and -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 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 -graded, the canonical inclusion is an isomorphism.

Proof. Note that for all and n > k, we have . Then

Lemma 28. Let and σ ∈ 𝔖_n. Then (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 -graded, . Under this identification, is the usual multiplication on .

Proof. Fix n ≥ 0. By Lemma 28, the isomorphism maps the subspace of symmetric elements in Tⁿ(V ^*) onto the space of symmetric linear maps . While the latter is isomorphic to by Remark 2, the former is isomorphic to via the linear map

This yields an isomorphism . With a similar reasoning as in the proof of Corollary 27, one obtains . It is then a straightforward computation to show that is indeed the usual multiplication on .

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 (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

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 and , respectively. A strict L∞ -algebra homomorphism is a degree preserving linear map f : L → L′ satisfying (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 , where for an antisymmetric linear map ω: 𝔤^⊗n → V , we define δω by

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 such that

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

As ↓ and ↑^⊗n are isomorphisms, this is equivalent to

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 for 1 ≤ k < ∞, all of degree one, such that (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 such that (30)

Proof. Combine the brackets in Proposition 33 to a single element 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 (31) for all k ≥ 1.

The equation (31) can be written more conveniently as (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 with d(1) = 0.

In this case, we also refer to the pair as an L_∞-algebra.

Proof. Let 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 is a coderivation of by Proposition 21, we have by Theorem 22 that d² = 0 if and only if

Corollary 37. If L is of finite type and -graded, an L_∞-structure on L is equivalent to a differential on the graded algebra . Explicitly, consider d_CE = −d^* for d as in Theorem 36.

Proof. See Remark 26 and note that as L is -graded, each of degree one vanishes on by degree reasons.

3.3 Weak homomorphisms

Let and be L_∞ -algebras, λ = pr_L[1]○d and . 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 .

Remark 39. By Proposition 21 and Theorem 22, a homomorphism of coaugmented coalgebras is a homomorphism of L_∞-algebras if and only if

Note that by Proposition 20, it makes sense to also refer to DGC homomorphisms as homomorphisms of L_∞-algebras.

From the dualised standpoint, we immediately get the following.

Proposition 40. Assume that L and L′ are -graded and of finite type. Then is a homomorphism of L_∞-algebras if and only if 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 is a weak one.

Proof. Observe that .

Lemma 42. A homomorphism of coalgebras preserves the exterior degree if and only if for a linear degree preserving map g: L[1] → L′[1].

Proof. Assume that is a homomorphism of coalgebras such that for all n and let . Then . Hence, by Proposition 18.

Proposition 43. Let be a (weak) L_∞-algebra homomorphism. Then f preserves the exterior degree if and only if 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 . 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 of degree one that satisfies (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 , where ∂ is a differential on V and 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 be a DGLA and be the corresponding linear degree one map . For a linear degree preserving map, the induced homomorphism of coalgebras is a homomorphism of L_∞-algebras if and only if (34)

This is the case if and only if the linear degree one map defined by satisfies (35)

Proof. The first part follows immediately from Remark 39 and the explicit construction of (see Proposition 18). It is straightforward to check that for homogeneous,

from which the second part then follows.

Proof of Theorem 45: As , a linear degree one map can be decomposed into linear degree one maps and ; the latter being equivalent to the choice of a degree one element . If we show that under this identification ρ satisfying (33) is equivalent to ∂² = 0 and satisfying (35), the assertion follows by Lemma 46. For homogeneous,

by cocommutativity of and

As , satisfying (35) is equivalent to (33) holding on . We also have

which completes the proof as .

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 is on 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 .

Remark 48. Let ρ be a representation of L on V and 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 , we obtain an L_∞-algebra homomorphism . The induced representation is given by and is called the representation dual to ρ.

• (2)

  Fix . Then is a differential on V [n] and is a DGLA homomorphism. The induced representation of L on V [n] is given by

4.2 Representations as coderivations

Observe that the map satisfies (36)

which makes into a left -comodule.

Definition 49. Let be of degree p. A coderivation of extending d′ is a linear map of degree p such that (37)

Proposition 50 ([6], Proposition 1.5.3, p. 31). Let be of degree p. There is a one-to-one correspondence between coderivations D of extending d′ and linear maps of degree p given by where is the projection of onto .

Proof. Let D be a coderivationof extending d′. As (id_S ⊗pr_V)(Δ_S ⊗id_V) = id_S ⊗id_V, we obtain from (37) that

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

Let conversely be of degree p. Using (36) and that d′ is a coderivation, we compute

which, combined, show that D := d ⊗id_V + (id_S ⊗ ρ)Δ_V is a coderivation of 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 extending d such that D² = 0.

Proof. It is a straightforward computation to check that is a coderivation of extending . By Proposition 50, D² = 0 if and only if ρ = pr_V ○ D satisfies

4.3 A first approach to L_∞-algebra cohomology

Assume now that the L_∞-algebra L is -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 , V ≅ V ^** and . Let d_CE = −d^* denote the differential on . The map

makes into a left -module. Similarly to Definition 49, we call a linear map of degree one a derivation of extending d_CE if

holds for all, 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 extending d_CE with . 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 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 where is identified with the linear map , . For homogeneous, one finds that D_CE f is then given by (38)

One could then simply define by (38), even if L and V do not meet our finiteness assumptions. Although there is a priori no reason for 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 . 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

Recall from Proposition 21 that is closed under the graded commutator. Together with Theorem 22, this induces a Lie bracket on . Its explicit formula is (39)

for homogeneous.

As L_∞-structures correspond to codifferentials with d(1) = 0 and elements in , it is only natural to restrict ourselves to the Lie subalgebra . Keeping the part corresponds to the framework of curved L∞-algebras, which are L_∞-algebras that also allow for a 0-ary bracket .

Remark 54. The same construction also makes into a graded Lie algebra. The decomposition implies that (40)

We can then consider spaces like and as subspaces of in the obvious way. The inclusion of into is then easily seen to preserve the Lie bracket.

Remark 55. In terms of the Lie bracket on , the condition (30) for a linear map of degree one to define an L_∞-algebra structure on L[1] becomes (41)

By Example 12 and Remark 54, this makes 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 and both by μ_S and Δ_S . This is justified, as they coincide on .

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 being (co)commutative. The similarity between (38) and (39) suggests to approach L_∞ -algebra cohomology using the Lie bracket on .

Proposition 56. Let be of degree one. Then ρ is a representation of L on V if and only if (42) where λ and ρ are considered as elements of .

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 . For x₁, …, x_n−1 ∈ L[1] and x_n ∈ V , a routine computation using Lemma 24 shows that

As again μ_S(ρ ⊗id_S)Δ_S = μ_S(id_S ⊗ ρ)Δ_S by (co)commutativity of , ρ satisfies (33) if and only if it satisfies (42).

Corollary 57. An element 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

Corollary 58. The subspace 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 .

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 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 into a DGLA. The space is then an abelian Lie subalgebra that is invariant under [λ + ρ, ⋅ ]. Explicitly, we have for homogeneous (43)

Definition 61. The map is called the L∞-coboundary operator. The cohomology of the cochain complex 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

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 , 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 . 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 , and 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 decomposes itself into linear maps

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 , the n-ary bracket decomposes into linear maps

These then correspond to cocycles in . 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).

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 . 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 .

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 (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 into linear degree one maps

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], . 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 , the condition for λ₁ + λ₂ + λ_m to define an L_∞-structure on L₁ ⊕ L₂ becomes (45)

The isomorphism allows for the following characterisation of semidirect sums.

Theorem 66. Let be of degree one. Then λ_m satisfies (45) if and only if the corresponding linear degree one map is a weak homomorphism of L_∞-algebras in the sense that it satisfies (35).

Proof. Note that is closed under [⋅, ⋅] and [λ₁ + λ₂, ⋅ ]. Therefore, (45) has only to be checked on . Let d₁ and d₂ denote the codifferentials on and , respectively. For and , 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 is a Lie subalgebra.

References