L∞-algebras and their cohomology

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.


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.

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 * e-mail: ben.reinhold@stud.uni-goettingen.de terms of symmetric brackets and codifferentials. The main references for this section are [3,4,6,7].

Graded vector spaces
A graded vector space is a vector space V together with a decomposition V ∼ = p∈Z V p for a family of vector spaces {V p } p∈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 of characteristic zero. We always denote by V and W graded vector spaces and by v 1 , . . . , v n ∈ V arbitrary homogeneous elements.
A linear map f : V → W is called homogeneous (of degree p) if there is p ∈ Z such that f (V n ) ⊂ W n+p for all n ∈ 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 p∈Z Hom p (V, W ). Elements in Hom 0 (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 allows us to define a grading on V ⊗ W by Proposition 1. Let f : V ⊗n → W be a symmetric linear map. There is a unique linear map ϕ : S n (V ) → W such that the following diagram commutes: where π S : V ⊗n → 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 ϕ : S n (V ) → 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 ) 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 symmetric by construction of S 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 ). An analogue of Proposition 1 holds for n V and induces an isomorphism between the subspace Hom(V ⊗n , W ) of all anti-symmetric maps and Hom( n V, 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 ) is isomorphic to the subspace of V ⊗n of all symmetric elements. Indeed, letting v 1 ∨ . . . ∨ v n denote the image of v 1 ⊗ . . . ⊗ v n under π S , the linear map ϕ : S n (V ) → V ⊗n , is well-defined and satisfies π S • ϕ = id S n (V ) and ϕ • π S = 1 n! σ∈Snε (σ). As the latter is a projection of V ⊗n onto said subspace, the claim follows. A similar statement clearly holds for n V . For n ∈ Z, we define 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 (↓ ⊗k ) −1 = (−1) k(k−1) 2 ↑ ⊗k as a consequence of (2).
There is then a degree preserving isomorphism Proof. Note that for the first part, we only have to check (4) on the generating subset {s 1 , . . . , s n−1 } ⊂ S n . This is an easy computation left to the reader. Let π A : V ⊗n → n V be the canonical projection. The linear maps π S • ↓ ⊗n : V ⊗n → S(V [1]), are then antisymmetric and symmetric, respectively. The induced linear maps between S(V [1]) and n V are then easily seen to be inverse to each other. As these maps are of degree −n and n, respectively, we obtain a degree

Corollary 4.
There is for each p ∈ Z a one-to-one correspondence between symmetric linear maps λ : 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 d 2 = 0. We then call the 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 : 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 H n (V ) = n-cocycles n-coboundaries the nth cohomology group.

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 general not 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) |a||b| 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 ∼ = n∈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 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≥0 V ⊗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 called the interior grading. On the other hand, T (V ) carries the grading given by T (V ) = n≥0 V ⊗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 We call S(V ) := 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 ⊗ v 2 + (−1) |v1||v2| v 2 ⊗ v 1 . We denote the multiplication in S(V ) and V by ∨ and ∧, respectively. Note that S(V ) and V also admit an exterior grading or grading by weight: 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 ) → A. Applying this to the linear map yields a homomorphism of graded unital algebras S(V ⊕ W ) → S(V ) ⊗ S(W ) that is easily seen to be an With a slight modification of the sign in (6), similar arguments show that (V ⊕ W ) ∼ = V ⊗ W . In particular, we have For a graded algebra A, a derivation of A of degree p is a linear map d : 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 p∈Z Der p (A). A differential on the graded algebra A is an element d ∈ Der(A) of degree one such that d 2 = 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.

Graded Lie algebras and unshuffle permutations
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) |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 The equation (10)  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 −1 i,n−i ⊂ S n . Using the antisymmetry of the Lie bracket, one can rewrite (9) as for all x 1 , x 2 , x 3 ∈ L homogeneous.
Lemma 15. Each element σ ∈ S n has for each i ∈ {0, . . . , n} a unique decomposition σ = τ (α, β), where τ ∈ 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.

Graded coalgebras
A (graded) coalgebra (C, ∆) is a graded vector space C together with a degree preserving linear map ∆ : 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 → k such that the diagram commutes. The map ε is then called the counit of C. If τ C,C • ∆ = ∆, then C is called cocommutative. A linear degree preserving map f : C → D between coalgebras (C, ∆ C ) and (D, ∆ D ) is called a homomorphism of coalgebras if 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, ∆) and n ∈ N, we define the iterated coproduct ∆ n : C → C ⊗(n+1) by ∆ 0 = id C and ∆ n = (∆ ⊗ id C ⊗ . . . ⊗ id C )∆ n−1 for n ≥ 1. It is convenient to then use Sweedler notation and to write ∆ n (x) ∈ C ⊗(n+1) for x ∈ C as In this notation, for example, the condition for C to be cocommutative becomes Lemma 16. Let (C, ∆ C ) be a coassociative coalgebra. Then for all p, q ∈ N, If (D, ∆ D ) is another coassociative coalgebra and if f : C → D is a coalgebra homomorphism, holds for all n ∈ N.

Coaugmented coalgebras
A coaugmented coalgebra (C, ∆, ε, u) is a counital coassociative coalgebra (C, ∆, ε) together with a homomorphism of counital coalgebras u : k → C. The coproduct on k is given by 1 k → 1 k ⊗ 1 k and its counit is the identity on k.
Denoting u(1 k ) ∈ C as 1, the conditions for u to be a homomorphism of counital coalgebras become ∆(1) = 1 ⊗ 1 and ε • u = id 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, ∆, ε, u), set C := ker(ε). We claim that C ∼ = C ⊕ k. Indeed, as ε • u = id k , the short exact sequence of graded vector spaces Using (13) and ε(x) = 0, one easily sees that for the last equality, note that is a coassociative coalgebra. Conversely, given a coassociative coalgebra (C, ∆), we define a coproduct on C := C ⊕ k by ∆(1) = 1 ⊗ 1 and This makes C into a coaugmented coalgebra; the counit and coaugmentation map are given by the projection C → k and the inclusion k → C, respectively. These constructions are clearly inverse to each other (up to isomorphism).
Let C and D be coaugmented coalgebras with counits ε 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 f : C → D. It is then easy to see that f is a homomorphism of coaugmented coalgebras if and only if f is a homomorphism of coalgebras. This yields a one-to-one correspondence between coalgebra homomorphisms C → 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, ∆) conilpotent if for all x ∈ C there is an n ∈ N such that ∆ n (x) = 0.

Examples of coalgebras
There is a coproduct ∆ A on T (V ) := n≥1 V ⊗n given by where we now denote the multiplication in T (V ) by concatenation to avoid ambiguities. This makes T (V ) into a coassociative graded coalgebra. The induced coaugmented coalgebra (T (V ), ∆ A ) is called the tensor coalgebra. Inductively, one finds which shows that T (V ) is conilpotent.
Proposition 17. Let (C, ∆) be a conilpotent coalgebra and f : C → V a linear degree preserving map. There is a unique homomorphism of coaugmented coalgebrasf : 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 coalgebrasf : For the uniqueness, assume that there is suchf . By Lemma 16, we havẽ Composing both sides with pr ⊗(n+1) V and noting that pr This shows thatf is completely determined by f and therefore unique. For the existence, consider the linear map ∞ n=0 ∆ n : C → T (C). This is well-defined as C is conilpotent. A straightforward computation using Lemma 16 and is easily seen to be also a coalgebra homomorphism,f : It is immediate that π • N = id S , where here and subsequently, we abbreviate S(V ) and S(V ) to S in subscripts. Using Lemma 15, we compute

See this equation above.
This shows that im(N ) ⊂ T (V ) is a subcoalgebra and induces a coproduct on S(V ) ∼ = im(N ). As a subcoalgebra of a coassociative coalgebra is clearly coassociative itself, S(V ) becomes a coaugmented coalgebra with the coproduct ∆ S : S(V ) → S(V ) ⊗ S(V ) given by Proposition 18. Let (C, ∆) be a cocommutative conilpotent coalgebra and f : C → V a degree preserving linear map. There is a unique homomorphism of coaugmented coalgebrasf : Proof. As in the proof of Proposition 17, it suffices to show that there is a unique homomorphism of coalgebras B. Reinhold: Emergent Scientist 3, 4 (2019) 7 We obtain an induced homomorphism of coalgebras with

Comodules and coderivations
Let (C, ∆) be a coassociative coalgebra. A left comodule over C is a graded vector space M together with a degree preserving linear map ∆ l : Similarly, a right comodule over C is a graded vector space M together with a degree preserving linear map If M is both a left and a right comodule over C and if the compatibility relation 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 We denote the vector space of all these maps by Coder p (M, C) and by Coder(M, C) the graded vector space p∈Z Coder p (M, C). Let (C, ∆ C ) and (D, ∆ D ) be coassociative coalgebras and f : D → C a coalgebra homomorphism. Then ∆ r := (id D ⊗ f )∆ D and ∆ l := (f ⊗ id D )∆ 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 f : D → C then makes D into a comodule over C. The following proposition relates elements in Coder(D, 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 d : Proof. Given a linear map d ∈ Hom(D, C), one easily It then suffices to show that each coderivation Let ε be the counit of C and µ k : k ⊗ k → k the multiplication on k.
From (13), (23) and the compatibility with the counit (24) it then follows that which shows that d(D) ⊂ C. Hence, d decomposes as For a coassociative coalgebra C, we call an element d ∈ Coder(C) of degree one with d 2 = 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.
Proof. Both parts of the proposition are straightforward computations which are left to the reader.
is then an isomorphism. Its inverse is given by The first part of Theorem 22 actually holds for a broader class of comodules over S(V ) (see for example [8], Lemma 2.4); the inverse formula d = µ S (λ ⊗ id M )∆ r then continues to hold for comodules M in which τ M,S • ∆ r = ∆ l .

See this equation above.
In the fourth equality, we shifted the summation index of the first sum and used Remark 23.
Proof of Theorem 22. Let d : D → S(V ) be a coderivation, that is Inductively, we then get For n ∈ N, let π n : T n (V ) → S n (V ) be the linear map defined by From S(V ) being a subcoalgebra of T (V ) and (17), it follows that π n+1 • pr As this holds for all n ∈ N and as pr k • 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 )∆ D is a coderivation with pr V • d = λ. While the latter holds by construction, we compute for x ∈ D homogeneous

See this equation next page
where we used Lemma 24 in the first and cocommutativity of D in the fifth equality.

Dual spaces
The graded vector space V * := Hom(V, k) is called the dual space of V . By degree reasons, (V * ) k = Hom k (V, k) = Hom(V −k , k) = (V −k ) * . For f ∈ Hom p (V, W ), the lin- Notions as Z <0 -graded or Z ≥0 -graded are defined accordingly. In the following, we denote V ⊗n as T n (V ) for better readability.

Proposition 25. If V is of finite type and if for all
has only finitely many non-trivial summands, then the canonical inclusion T n (V * ) → T n (V ) * is an isomorphism.
Proof. It is well-known that for finite-dimensional (ungraded) vector spaces V 1 , . . . , V n , the canonical inclu- Remark 26. If V is of finite type and Z ≤0 -graded, V * is also of finite type and Z ≥0 -graded and they both satisfy the hypothesis of Proposition 25. It is then easy to see that ∆ : V → V ⊗ V makes V into a graded coassociative/cocommutative coalgebra if and only if ∆ * : V * ⊗ V * ∼ = (V ⊗ V ) * → V * makes V * into an associative/commutative algebra. A linear map d : V → V is then a coderivation of (V, ∆) if and only if −d * is a derivation of (V * , ∆ * ). 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 -graded, the canonical inclusion T (V * ) → T (V ) * is an isomorphism.
Proof. Note that for all k ∈ Z and n > k, we have T n (V * ) −k = 0. Then Proof. It suffices to show this for σ = s 1 , . . . , s n−1 , in which case it is an easy computation.
Proposition 29. If V is of finite type and Z <0 -graded, S(V ) * ∼ = S(V * ). Under this identification, ∆ * S is the usual multiplication on S(V * ).
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 holds for all n ≥ 1 and x 1 , . . . , x n ∈ L homogeneous. We then call the set {l k | 1 ≤ k < ∞} an L ∞ -structure on L.

See this equation above
for all x 1 , x 2 , x 3 ∈ L homogeneous. While the first two equations may be summarized by saying that l 1 is a differential on the (non-associative) graded algebra (L, l 2 ), a comparison with (11) shows that the third one describes the defect of the Jacobi identity in (L, l 2 ). 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, l 2 ) is an (ungraded) Lie algebra.
Definition 31. Let L and L be L ∞ -algebras with L ∞ -structures {l k } k∈N and {l k } k∈N , respectively. A strict L ∞ -algebra homomorphism is a degree preserving linear map f : L → L satisfying 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.

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 n≥0 Hom( n g, V ), δ , where for an antisymmetric linear map ω : g ⊗n → V , we define δω by for x 1 , . . . , x n+1 ∈ g. In the sums above, elements withâ re 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 l 1 = 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 = L 0 ⊕ L −n with l 1 = 0, all brackets exept for l 2 and l n+2 have to vanish by degree reasons. Also, l 2 has to vanish on 2 L −n and l n+2 can only be non-trivial on n+2 L 0 with image in L −n . Using (8), we can decompose l 2 into linear maps [·, ·] : 2 L 0 → L 0 and ρ : L 0 ⊗ L −n → L −n . It is then a matter of computation to show that l 2 and l n+2 satisfying (27) amounts to (L 0 , [·, ·]) being a Lie algebra, ρ being a representation of L 0 on L −n and l n+2 being a cocycle.

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 : 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
Corollary 34. An L ∞ -structure on the graded vector space L is equivalent to a linear degree one map λ : S(L[1]) → L [1] such that Proof. Combine the brackets in Proposition 33 to a single element λ = k λ k ∈ Hom(S(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.
for all k ≥ 1.
The equation (31) can be written more conveniently as We then also refer to g as a strict L ∞ -algebra homomorphism.
Note that by Proposition 20, it makes sense to also refer to DGC homomorphisms S(L [1]) → S(L [1]) as homomorphisms of L ∞ -algebras.
From the dualised standpoint, we immediately get the following. From this it follows for example that all (weak) L ∞algebra homomorphisms between Lie algebras are induced by Lie algebra homomorphisms.

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] * ) ⊗ 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.

Theorem 45.
There is a one-to-one correspondence between representations of L on V and pairs (∂,f ), where ∂ is a differential on V andf : S(L[1]) → S(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).

This is the case if and only if the linear degree one map
Proof. The first part follows immediately from Remark 39 and the explicit construction off (see Proposition 18). It is straightforward to check that for x ∈ S(L [1]) homogeneous, f (d(x)) = (−1) |x| ↓ρ(d(x)), , from which the second part then follows.
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 : 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.
Remark 48. Let ρ be a representation of L on V and (∂,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 withf , we obtain an L ∞ -algebra homomorphism S(L[1]) → S(gl(V * ) [1]). The induced representation is given by and is called the representation dual to ρ. (2) Fix n ∈ Z. Then (−1) n ↓ n • ∂ • ↑ n is a differential on V [n] and is a DGLA homomorphism. The induced representation of L on V [n] is given by

Representations as coderivations
Observe that the map This shows that D is completely determined by pr V • D.
Let conversely ρ ∈ Hom(S(L [1]) ⊗ V, V ) be of degree p. Using (36) and that d is a coderivation, we compute

A first approach to L ∞ -algebra cohomology
Assume now that the L ∞ -algebra L is Z ≤0 -graded and of finite type and that V is either finite-dimensional or of finite type and trivial in the negative degrees.
holds for all ξ, η ∈ S(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.

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.
One could then simply define D CE : Hom(S(L[1]), V ) → Hom(S(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 2 CE = 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(S(L [1]), S(L[1])) ∼ = Hom(S(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.  By abuse of notation, we now denote the (co)products on S(L [1]) and S(L[1] ⊕ V ) both by µ S and ∆ S . This is justified, as they coincide on S(L [1] where λ and ρ are considered as elements of Hom(S( Proof. Note that ρ • µ S (ρ ⊗ id S )∆ S and ρ • µ S (ρ ⊗ id S )∆ S are only possibly nonzero on S(L[1]) ⊗ V . For x 1 , . . . , x n−1 ∈ L[1] and x n ∈ V , a routine computation using Lemma 24 shows that Proof. We have 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 : 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 L n−1 ⊗ V for each n ≥ 1. Representations of L ∞ -algebras are often defined in terms of these equations, see for example (  ). This is the case discussed in [4].

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

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 ∈ S(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 ∈ S(I [1]).
Given such an exact sequence (44), the graded subspace L 2 ∼ = ker(p) ⊂ L is an ideal and p induces a strict isomorphism L/L 2 ∼ = L 1 of L ∞ -algebras.

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 L 2 → L → L 1 is called abelian if L 2 is abelian. The

Semidirect sums
An L ∞ -algebra ((L 1 ⊕ L 2 )[1], λ) is said to be a semidirect sum of the L ∞ -algebras (L 1 [1], λ 1 ) and (L 2 [1], λ 2 ) if the canonical sequence L 2 → L 1 ⊕ L 2 → L 1 is an L ∞ -algebra extension and if the canonical map L 1 → L 1 ⊕ L 2 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 L 1 and L 2 is therefore characterised by λ m . Note that L 1 ⊂ L 1 ⊕ L 2 is an ideal if and only if λ m = 0. In this case, L 1 ⊕ L 2 carries the L ∞ -structure λ 1 + λ 2 and is called the direct sum of L 1 and L 2 .