Shifted L∞ Bialgebras

Masterarbeit

Shifted L∞ bialgebras

Malte Dehling

September 19, 2011 Contents

Introduction 2

1 Preliminaries 4 1.1 Gradedvectorspaces...... 4 1.2 The graded tensor algebras ...... 5 1.3 The graded tensor coalgebras ...... 7

2 Lie bialgebras 9 2.1 Liealgebras ...... 9 2.2 Liebialgebras ...... 10 2.3 Manintriples ...... 11

3 L∞ algebras 13

3.1 L∞ algebras via brackets ...... 13

3.2 L∞ algebras via (co)derivations ...... 16

3.3 L∞ algebras via the big bracket ...... 18

4 L∞[α] bialgebras 24

4.1 L∞[α] bialgebras via the big bracket ...... 24

4.2 Manin L∞[α] triples...... 25 4.3 Representations up to homotopy ...... 28

4.4 L∞ algebra cohomology ...... 31

Bibliography 33

Eigenständigkeitserklärung 36

1 Introduction

In this thesis we construct certain categorifications of Lie bialgebras. A Lie bialgebra is a pair of Lie algebra structures on dual vector spaces (g, g∗). The two structures are required to satisfy a certain compatibility condition which can be expressed in different ways. Lie bialgebras appear naturally as the infinitesimal versions of Poisson Lie groups. The Poisson structure gives a Lie algebra structure on g∗ and the compatibility comes from the fact that the Lie group multiplication is a Poisson map. The precise statement is due to Drinfel’d (Dri88): The category of connected simply connected Poisson Lie groups is equivalent to the category of finite-dimensional Lie bialgebras. Since Lie bialgebras and Poisson Lie groups can be regarded as classical limits of quantum groups, we hope that our categorification of Lie bialgebras may help to better understand categorification of quantum groups as was re- cently done in (Lau10), (KL09), and (KL11), although, we must admit that at first sight these categorifications appear to be quite different. A starting point for categorification of Lie bialgebras is categorification of Lie algebras. This has been done in (BC04). Their Lie 2-algebras are 2-term

L∞ algebras. In (Kra07) one finds a definition for an L∞ bialgebra L, however, while there is an L∞ algebra structure on L, in her setting there is no such structure on L∗. We solve this problem by introducing a simple shift in degree. Then the problem remains to identify a good set of compatibility relations for the L∞ structures on L and L′ — its shifted dual. For this we use a very powerful tool due to Kosmann-Schwarzbach: the big bracket. For our Lie k-bialgebras we give a corresponding notion of Manin triple and prove equivalence. Finally, we give several equivalent definitions of L∞ modules and define L∞ cohomology with coefficients in such modules, extending the work in (Pen95) and (Pen01).

2 We expect to ﬁnd a deﬁnition for Lie k-bialgebras (with strict compatibility) via

L∞ cocycles. Let us point out some related work in recent preprints.

(1) In (BSZ11), much work was done on the case of (strict) Lie 2-bialgebras. Examples are constructed by solving higher classical Yang-Baxter equa- tions. We expect some similar results to hold in the non-strict case. This is work in progress.

(2) In (CSX11), weak Lie 2-bialgebras are deﬁned using a similar big bracket construction. These weak Lie 2-bialgebras are the same as our Lie 2- bialgebras with strict compatibility. The main result is that for the strict case, Lie bialgebra crossed modules are in bijection with Lie 2-bialgebras.

Finally, I wish to thank Chenchang Zhu for explaining so many things to me and for her encouragement to ﬁnish this thesis.

3 Chapter 1

Preliminaries

1.1 Graded vector spaces

Z A -graded vector space is a vector space V = k∈Z Vk, i.e. any vector v ∈ V has a unique decomposition v = ∑ vk with vkL∈ Vk. We call vk the homogeneous component of degree k. Whenever we use a basis for such a graded vector space, we will choose it to be homogeneous by choosing bases for Vk in- dividually. Graded vector spaces form a category with morphisms the degree- preserving maps, i.e.

Hom(V, W)= { f : V → W linear | f (Vk) ⊂ Wk}.

We view an ungraded vector space V as a graded vector space with trivial decomposition V0 = V, Vk = {0} for k 6= 0. In fact, this deﬁnes an embedding of ungraded vector spaces in Z-graded vector spaces. For graded vector spaces V, W, we deﬁne their tensor product by

V ⊗ W = Vi ⊗ Wj . Z   kM∈ k=Mi+j   Often we will be interested not just in degree-preserving maps but in homogeneous maps of deg p. We denote these internal homs by

p Hom (V, W)= { f : V → W linear | f (Vk) ⊂ Wk+p}.

For such maps f ∈ Hom•(V, W), g ∈ Hom•(V′, W′), their tensor product is

4 deﬁned by ( f ⊗ g)(v ⊗ v′)=(−1)|g||v| f (v) ⊗ g(v′), where |g|, |v| denote the degrees of g resp. v. For composition we have a similar rule,

| f2||g1| ( f1 ⊗ f2) ◦ (g1 ⊗ g2)=(−1) f1 ◦ g1 ⊗ f2 ◦ g2.

By this rule, for f , g invertible, ( f ⊗ g)−1 =(−1)| f ||g| f −1 ⊗ g−1. We deﬁne a functor [n] for n ∈ Z. This functor maps a graded vector space to a copy of itself shifted down by n, i.e. (V[n])k = Vn+k, and similarly for maps. Clearly, V and V[n] are isomorphic by some degree ±n map. We denote this map by

↓ n : V → V[n], v 7→↓ nv = v[n], and its inverse (for W = V[n]) by

↑ n : W → W[−n].

When n = 1, we will write just ↓, ↑. ∗ ∗ For a graded vector space V, we deﬁne its dual by (V )k = (V−k) , thus V∗ =∼ Hom•(V, K). Note that V[n]∗ = V∗[−n] canonically.

1.2 The graded tensor algebras

We have already seen the deﬁnition for the tensor product of Z-graded vector spaces. In the same way, one can deﬁne the grading on V⊗k = V ⊗ ... ⊗ V, the k-fold tensor product of V by

⊗k (V )l = (Vi1 ⊗ ... ⊗ Vik ). l=i1M+...+ik

⊗k The tensor algebra is the sum T(V)= k∈N V with the concatenation product. It is the free associative Z-gradedL algebra generated by V. We want to think about symmetric and anti-symmetric elements of T(V) in some graded sense. The general idea here is that two commutative “things” x, y of degrees |x|, |y| satisfy xy = (−1)|x||y|yx, while for anti-commutative

5 things, an additional minus sign comes into play. The terminology used here is consistent with the one used for ungraded things if one thinks of ungraded things as being graded in degree zero.

Let us denote by Sk the group of all bijections of the set {1,..., k}. The group Sk is generated by permutations σi exchanging only i and i + 1, i.e.

σi : i 7→ i + 1, i + 1 7→ i, j 7→ j ∀j 6= i, i + 1

The σi are subject to the relations

2 > σi = 1, σiσi+1σi = σi+1σiσi+1, σiσj = σjσi |i − j| 1.

k We will now deﬁne two actions of Sk on T (V). For generators σi and homogeneous elements x1,..., xk ∈ V, we deﬁne the even action

ǫ(σi)(x1 ⊗ ... ⊗ xk)=

|xi||xi+1| (−1) x1 ⊗ ... ⊗ xi−1 ⊗ xi+1 ⊗ xi ⊗ xi+2 ⊗ ... ⊗ xk, and the odd action

χ(σi)(x1 ⊗ ... ⊗ xk)=

|xi||xi+1| − (−1) x1 ⊗ ... ⊗ xi−1 ⊗ xi+1 ⊗ xi ⊗ xi+2 ⊗ ... ⊗ xk,

One easily veriﬁes compatability with the relations on the generators. We will call elements of T(V) which are invariant under ǫ resp. χ graded symmetric resp. anti-symmetric. Often we will denote by ǫ(σ; x1,..., xk) or ǫ(σ) just the sign for the permutation, i.e.

ǫ(σ)(x1 ⊗ ... ⊗ xk)= ǫ(σ)xσ(1) ⊗ ... ⊗ xσ(k), and similarly for χ. We then call ǫ(σ) resp. χ(σ) the graded commutative resp. anti-commutative Koszul sign. We have already deﬁned which elements of T(V) we regard as symmetric resp. anti-symmetric in the graded sense. We now use the Koszul signs to deﬁne the graded (anti)symmetrization operators S, A : T(V) → T(V)

1 S : x ⊗ ... ⊗ x 7→ ∑ ǫ(σ; x ,..., x ) x ⊗ ... ⊗ x 1 k k! 1 k σ(1) σ(k) σ∈Sk

6 and

1 A : x ⊗ ... ⊗ x 7→ ∑ χ(σ; x ,..., x ) x ⊗ ... ⊗ x . 1 k k! 1 k σ(1) σ(k) σ∈Sk These are projection operators and the image S(V) = S(T(V)) of S consists precisely of the graded symmetric tensors, while (V) = A(T(V)) are the graded anti-symmetric tensors. V In general, the tensor product of two (anti)symmetric elements is not (anti)- symmetric. However, we can deﬁne

∨ : S(V) ⊗S(V) →S(V) x ⊗ y 7→ x ∨ y = S(x ⊗ y),

∧ : (V) ⊗ (V) → (V) x ⊗ y 7→ x ∧ y = A(x ⊗ y). ^ ^ ^ With this deﬁnition, (S(V), ∨) and ( (V), ∧) become associative algebras. The shifting isomorphisms ↑, ↓mapV even elements to odd elements and vice versa. This ofcourse exchanges the meaning of symmetric and anti-symmetric. The diagram

χ(σ) / Tk(V) / Tk(V)

↓⊗k ↓⊗k (1.1) ǫ(σ) / Tk(V[1]) / Tk(V[1]) is commutative, since

⊗k ǫ(σi) ↓ (x1 ⊗ ... ⊗ xk)

k ∑ (k−i)|xi| = ǫ(σi) (−1) i=1 x1[1] ⊗ ... ⊗ xk[1]

k |xi[1]||xi+1[1]| ∑ (k−i)|xi| =(−1) (−1) i=1 x1[1] ⊗ ... ⊗ xi+1[1] ⊗ xi[1] ⊗ ... ⊗ xk[1]

|xi||xi+1| ⊗k = −(−1) ↓ (x1 ⊗ ... ⊗ xi+1 ⊗ xi ⊗ ... ⊗ xk) ⊗k =↓ (χ(σi)(x1 ⊗ ... ⊗ xk))

k ∼ k for all generators σi. This deﬁnes an isomorphism (V)[k] = S (V[1]), some- ⊗ − k(k−1) ⊗ times called the décalage isomorphism. Note that (V↓ k) 1 =(−1) 2 ↑ k.

1.3 The graded tensor coalgebras

In the previous section we have defined the algebra structures on S(V) and −1 (V). We will also need their coalgebra structures. Let us denote by Shi,n−i the V 7 set of (i, n − i)-unshuffles, i.e. the permutations of {1,..., n} satisfying σ(1) < ... < σ(i) and σ(i + 1) < ... < σ(n). Think of an unshuffle permutation as splitting a single pile of cards into two. We use these permutations to define the unshuffle coproduct on (V), V ∆(x1 ∧ ... ∧ xk)= ∑ ∑ χ(σ)(xσ(1) ∧ ... ∧ xσ(i)) ⊗ (xσ(i+1) ∧ ... ∧ xσ(k)). i −1 σ∈Shi,k−i

With this coproduct, (V) becomes a coassociative coalgebra. ( (V), ∧, ∆) is in fact a biassociativeV bialgebra. Similarly, we deﬁne the coalgebraV structure on S(V) by

∆(x1 ∨ ... ∨ xk)= ∑ ∑ ǫ(σ)(xσ(1) ∨ ... ∨ xσ(i)) ⊗ (xσ(i+1) ∨ ... ∨ xσ(k)). i −1 σ∈Shi,k−i

8 Chapter 2

Lie bialgebras

Lie bialgebras are Lie algebras such that the dual of the underlying linear space is again a Lie algebra. The two Lie brackets are required to satisfy a certain compatibility condition. To ﬁx notation we start by recalling some basic deﬁni- tions.

2.1 Lie algebras

Deﬁnition 1. A Lie algebra (g, [·]) consists of

(1) a K-vecter space g, and

(2) a linear map [·] : g ∧ g → g, such that the Jacobi identity

[[x ∧ y] ∧ z]+[[y ∧ z] ∧ x]+[[z ∧ x] ∧ y]= 0 holds.

Example 2. (1) For any associative algebra A, the commutator deﬁned by [a ∧ b]= ab − ba will give (A, [·]) the structure of a Lie algebra. The proof of the Jacobi identity will use associativity.

(2) For any vector space E, the space End(E) of linear endomorphisms with composition is associative. Thus gl(E)= End(E) equipped with the commutator bracket [A ∧ B]= A ◦ B − B ◦ A is a Lie algebra according to the previous example.

9 Deﬁnition 3. A morphism of Lie algebras

f : (g, [·]) → (g′, [·]′) is a linear map f : g → g′, s.t. f ([x ∧ y])=[ f (x) ∧ f (y)]′ for all x, y ∈ g. A representation ρ of a Lie algebra g on some vector space E is a morphism ρ : g → gl(E) of Lie algebras.

The examples relevant to our discussion of Lie bialgebras are

(1) the adjoint representation

ad: g → gl( g) x 7→ adx, (2.1) ^ where adx is given on g by adx : y 7→ [x ∧ y] and then extended to a derivation of g. V (2) The coadjoint representation ad∗ : g → gl(g∗) deﬁned using the dual pairing h·|·i : g∗ ⊗ g → K:

∗ hξ|adxyi = −hadxξ|yi.

2.2 Lie bialgebras

Deﬁnition 4. Let (g, [·]) be Lie algebra and γ : g → g ∧ g such that [·]∗ = γ∗ : g∗ ∧ g∗ → g∗ deﬁnes a Lie algebra (g∗, [·]∗). Then (g, [·], γ) is a Lie bialgebra, provided

• the Lie brackets are compatible in the sense that γ : g → g ∧ g is a 1- cocycle w.r.t. the adjoint representation, i.e.

γ([x, y])=(adx ⊗ 1 + 1 ⊗ adx)γ(y) − (ady ⊗ 1 + 1 ⊗ ady)γ(x).

For generalities on Lie algebra cohomology, see (HN91). Classically, e.g. in (Dri83), the compatibility condition was expressed in terms of the structure coefficients of the Lie (co)bracket. For a given basis {ei} i ∗ i i of g, denote by {e } the dual basis of g defined by the dual pairing he |eji = δj. k i j ∗ Then the structure coefficient are defined by [ei ∧ ej] = λijek, resp. [e ∧ e ] = ij k γk e . The compatibility condition now translates to

k ij i jα j iα i jα j iα λrsγk = λαrγs − λαrγs − λαsγr + λαsγr . (2.2)

10 From the formal symmetry of (2.2) it is immediate, that the concept of Lie bialgebra is self-dual.

Example 5. The Lie algebra of the special linear group SL(2, C) is denoted by sl(2, C). As a vector space it is the subspace of 2 by 2 matrices with vanishing trace. We choose a basis

1 0 0 1 0 0 H = , X = , Y = . 0 −1 0 0 1 0       These satisfy the following commutator relations:

[X ∧ Y]= H, [H ∧ X]= 2X, [H ∧ Y]= 2Y.

We can make sl(2, C) a Lie bialgebra by

γ(H)= 0, γ(X)= X ∧ H, γ(Y)= Y ∧ H.

The compatibility relation is now easy to check, e.g. for X, Y it is

(adX ⊗ 1 + 1 ⊗ adX)γ(Y) − (adY ⊗ 1 + 1 ⊗ adY)γ(X) =[X ∧ Y] ∧ H + Y ∧ [X ∧ H] − [Y ∧ X] ∧ H − X ∧ [Y ∧ H]

= H ∧ H − 2Y ∧ X + H ∧ H − 2X ∧ Y

= 0 = γ(H)= γ([X ∧ Y]).

2.3 Manin triples

To any Lie bialgebra (g, [·], γ), we can construct a Lie bracket on the space d = g ⊕ g∗. It is deﬁned by

[x ∧ y]d =[x ∧ y]

 ∗ ∗ ∗ [x ∧ ξ]d = ad ξ − ad x ∀x, y ∈ g, ξ, η ∈ g . (2.3)  x ξ  [ξ ∧ η]d =[ξ, ∧η]∗   We will call(g, [·]d) the double of the Lie bialgebra (g, [·, ·], γ). Note that d comes with a symmetric bilinear form deﬁned by

hx + ξ|y + ηid = hξ|yi + hη|xi.

11 With this deﬁnition, g and g∗ are isotropic Lie subalgebras, i.e. hg|gi = 0 = hg∗|g∗i. The Lie bracket [·] is invariant w.r.t. this bilinear form. The double of a Lie bialgebra is an example of what is called a Manin triple.

Deﬁnition 6. A Manin triple (d; g, g′) consists of a Lie algebra d, equipped with an invariant, non-degenerate, symmetric bilinear form, and two comple- mentary isotropic Lie subalgebras g, g′ of d.

In fact, every Manin triple arises as the double of some Lie bialgebra.

Theorem 1. There is a one-to-one correspondence between ﬁnite-dimensional Lie bialgebras and ﬁnite-dimensional Manin triples.

For a more detailed exposition and a proof of this classical case see e.g. (Kos97).

12 Chapter 3

L∞ algebras

L∞ algebras, also called strongly homotopy Lie algebras, are the Lie counter- part to the A∞ or strongly homotopy associative algebras of (Sta63a, Sta63b).

The original references for L∞ algebras are (LS93) and (LM95).

We will give ﬁve deﬁnitions of L∞ algebras, and prove equivalence under our general assumptions

(1) K, the ground ﬁeld, is of characteristic 0, and

(2) all (graded) vector spaces are (degree-wise) ﬁnite-dimensional.

3.1 L∞ algebras via brackets

We will give two definitions of L∞ algebras via higher brackets. The two are equivalent by a shift of the underlying graded vector space. The first definition is from (LM95).

Deﬁnition 7. An L∞ algebra (g, [·]) consists of

(1) a Z≤0-graded vector space g, and

(2) a collection of linear maps [·] : k g → g of degree 2 − k called k-brackets. V We require these k-brackets to satisfy the following generalized Jacobi iden-

13 tity: for all n,

i(j−1) ∑ (−1) ∑ χ(σ)[[xσ(1) ∧ ... ∧ xσ(i)] ∧ xσ(i+1) ∧ ... ∧ xσ(n)]= 0. i+j=n+1 −1 σ∈Shi,n−i (3.1)

The advantage of this deﬁnition is, that it immediately generalizes several others, including that of a Lie algebra, if one thinks of ungraded vector spaces as Z-graded solely in degree zero.

Example 8. (1) As mentioned above, for g = g0 ungraded, all brackets but the 2-bracket vanish for degree reasons. The 2-bracket is then required to satisfy

∑ χ(σ)[[xσ(1) ∧ xσ(2)] ∧ xσ(3)]= 0 σ −1 where σ ranges over Sh2,1 . This condition is the usual Jacobi identity.

(2) More generally, L∞ algebras where g = g−k+1 ⊕ ... ⊕ g0 are called k-term

L∞ algebras or Lie k-algebras. See also (BC04).

(3) Instead of restricting g to only some degrees, we can require some of the brackets to vanish. Requiring all brackets but the 1-bracket to vanish, condition (3.1) becomes [[x]] = 0. Thus g is a differential graded vector space in degrees Z≤0.

(4) Allowing for a 1-bracket and a 2-bracket, condition (3.1) will give three conditions,

(i) [[x]] = 0,

(ii) [[x ∧ y]] = [[x] ∧ y]+(−1)|x|[x ∧ [y]],

(iii) the Jacobi identity as above.

This is the deﬁnition of a differential Z≤0-graded Lie algebra.

Using the décalage isomorphism, we can rewrite deﬁnition 7 in the following form.

Deﬁnition 9. An L∞ algebra (L, λ) consists of

< (1) a Z 0-graded vector space L, and

14 (2) a linear map λ : S(L) → L of degree 1, s.t. for all n, n ∑ ∑ ǫ(σ)λ(λ(xσ(1) ∨ ... ∨ xσ(i)) ∨ xσ(i+1) ∨ ... ∨ xσ(n))= 0. (3.2) i=1 −1 σ∈Shi,n−i The equivalence of this deﬁnition to the previous one is commented on in (Vor05) and (CS08), however the explicit calculations are missing there.

Lemma 2. Deﬁnitions 7 and 9 are equivalent.

Proof. Let (g, [·]) be an L∞ algebra as in definition 7. We define L = g[1] =↓ g ( − ) k k 1 ⊗k and linear maps λk = (−1) 2 ↓◦[·]◦ ↑ . This composition is a symmetric map of degree 1 and it remains to check that for λ = ∑k λk, the tuple (L, λ) satisfies condition (3.2).

Let ↓ x1,..., ↓ xn be elements of ↓g = L, then using our deﬁnition for λk in

∑ ∑ ǫ(σ)λj(λi(↓ xσ(1) ∨ ... ∨↓ xσ(i))∨↓ xσ(i+1) ∨ ... ∨↓ xσ(n)) i+j=n+1 −1 σ∈Shi,n−i gives

= ∑ ∑ ±↓ [[xσ(1) ∧ ... ∧ xσ(i)] ∧ xσ(i+1) ∧ ... ∧ xσ(n)], i+j=n+1 −1 σ∈Shi,n−i where ± denotes the following sign

i j ∑ = (i−k)|xσ(k)| (j−1)|[xσ(1)∧...∧xσ(i)]|+∑ (j−k)|xσ(i+k−1)| ǫ(σ; ↓ x1,..., ↓ xn)(−1) k 1 (−1) k=2 ∑i (i−k)|x | ∑n (n−k)|x | (j− ) ∑i |x |+(j− )( −i) =ǫ(σ)(−1) k=1 σ(k) (−1) k=i+1 σ(k) (−1) 1 k=1 σ(k) 1 2

n i(j−1) ∑ = (n−k)|xσ(k)| =(−1) ǫ(σ; ↓ x1,..., ↓ xn)(−1) k 1 n i(j−1) ∑ (n−k)|xk| =(−1) χ(σ; x1,..., xn)(−1) k=1 .

Here, we have used commutativity of (1.1). Since the last factor does not de- pend on i, j, or σ, this proves equivalence of conditions (3.1) and (3.2) under the isomorphism.

The inverse construction is given by decomposing λ = ∑ λk, where λk : k ⊗k S (L) → L, deﬁning g =↑ L and the k-brackets by [·]=↑◦λk◦↓ .

′ ′ Deﬁnition 10. A strict morphism of L∞ algebras f : (L, λ) → (L , λ ) is a linear map f : L → L′ of degree 0 satisfying

′ ⊗k f ◦ λk = λk ◦ f , ∀k.

15 This deﬁnition is too strict for our purposes. We will give another (equivalent) deﬁnition of L∞ algebras which comes naturally with a weaker notion of morphism.

3.2 L∞ algebras via (co)derivations

It is well-known that a derivation is determined by its value on generators. In this section we will use the a dual statement, Hom(S(L), L) =∼ Coder(S(L)), to give a third deﬁnition for L∞ algebras. Dualizing will lead to a fourth deﬁ- nition.

Deﬁnition 11. An L∞ algebra (S(L), d) consists of

< (1) a Z 0-graded vector space L, and

(2) a coderivation d of degree +1 of the coalgebra S(L), s.t. d is a codifferential, i.e. d2 = 0.

Lemma 3. Deﬁnitions 9 and 11 are equivalent.

A similar proof is split over (LS93) and (LM95).

Proof. Let (L, λ) be an L∞ algebra as in deﬁnition 9. As before, we decompose k λ = ∑ λk. Now we extend these λk : S (L) → L by coderivation of the unshuf- ﬂe coproduct,

dk(x1 ∨ ... ∨ xn)=

0 n < k  ∑ −1 ǫ(σ)λk(xσ(1) ∨ ... ∨ xσ(k)) ∨ xσ(k+1) ∨ ... ∨ xσ(n) n ≥ k,  σ∈Shk,n−k  and deﬁne d = ∑ dk. Then condition (3.2) is exactly λ ◦ d = 0, which is equivalent to d2 = 0. Under our assumption that L be ﬁnite-dimensional, this is clear by duality: a derivation of an algebra is a differential, iff d2 = 0 holds on generators.

From an L∞ algebra (S(L), d), one can recover the map λ by projection,

λ = projL ◦ d.

By duality, one can equivalently give the following deﬁnition.

16 ∗ Deﬁnition 12. An L∞ algebra (S(L ), dCE) consists of

< (1) a Z 0-graded vector space L, and

∗ (2) a derivation dCE of degree +1 of the algebra S(L ),

2 s.t. dCE is a differential, i.e. dCE = 0.

For any L∞ algebra (S(L), d) as in deﬁnition 11, we denote by CE(L) = ∗ S(L ) its Chevalley-Eilenberg algebra. It is an L∞ algebra as deﬁned above ∗ with differential dCE(L) = d .

With this deﬁnition of L∞ algebra, the natural notion of morphism is a morphism of (co)differential (co)algebras. Note that this is different from the strict morphisms deﬁned previously.

Deﬁnition 13. A morphism of L∞ algebras is a morphism of codifferential coalgebras f : (S(L), d) → (S(L′), d′).

Example 14. For a Lie 2-algebra, L = L−2 ⊕ L−1. Then

1 S (L)= L−2 ⊕ L−1 2 2 2 S (L)= S (L−2) ⊕ (L−2 ⊗ L−1) ⊕ S (L−1) 3 2 3 S (L)= ... ⊕ L−2 ⊗S (L−1) ⊕ S (L−1), thus for degree reasons a coalgebra morphism f : S(L) → S(L′) is deﬁned by three maps:

′ f−1 : L−1 → L−1 ′ f−2 : L−2 → L−2 2 ′ f−1,−1 : S (L−1) → L−2

′ These are required to satisfy f ◦ d = d ◦ f which leads to (for x, y, z ∈ L−1, h ∈ L−2):

′ (1) f−1(λ1(h)) = λ1( f−2(h)),

′ ′ (2) f−1(λ2(x ∨ y)) = λ1( f−1,−1(x ∨ y))+ λ2( f−1(x) ∨ f−1(y)),

′ (3) f−2(λ2(x ∨ h)) − f−1,−1(x ∨ λ1(h)) = λ2( f−1(x) ∨ f−2(h)),

17 (4) f−2(λ3(x ∨ y ∨ z))+ f−1,−1(λ2(x ∨ y) ∨ z)+ f−1,−1(λ2(y ∨ z) ∨ x)+ ′ f−1,−1(λ2(z ∨ x) ∨ y)= λ3( f−1(x) ∨ f−1(y) ∨ f−1(z)) ′ ′ + λ2( f−1,−1(x ∨ y) ∨ f−1(z))+ λ2( f−1,−1(y ∨ z) ∨ f−1(x)) ′ + λ2( f−1,−1(z ∨ x) ∨ f−1(y)) .

For strict morphisms, f−1,−1 = 0 and the conditions simplify considerably.

Note that even for strict Lie 2-algebras (λ3 = 0) their strict morphisms are different from these more general morphisms.

3.3 L∞ algebras via the big bracket

Let V be a graded vector space. We deﬁne

B = S(V[a]∗ ⊕ V[b])

=∼ S(V[a]∗) ⊗S(V[b]) =∼ S(V∗[−a]) ⊗S(V[b]).

An element

u =ξ1[−a] ∨ ... ∨ ξk[−a] ∨ y1[b] ∨ ... ∨ yl[b] of B is then of (total) degree

k l k l |u| = ∑ |ξi[−a]| + ∑ |yi[b]| = ∑(|ξi| + a)+ ∑(|yi| − b) i=1 i=1 i=1 i=1 k l = ∑ |ξi| + ∑ |yi| +(ka − lb) i=1 i=1 With respect to this grading, the symmetric product ∨ is of degree 0. The dual pairing gives a map h·|·i : (V[a]∗ ⊕ V[b]) ⊗ (V[a]∗ ⊕ V[b]) → K, by

hξ[−a]+ x[b]|η[−a]+ y[b]i = ξ(y) − (−1)(|η[−a]|+α)(|x[b]|+α)η(x), which is of degree α = b − a. We extend this pairing to B by

(1) graded anti-symmetry

hhu|vii = − (−1)(|u|+α)(|v|+α)hhv|uii,

(2) bilinearity, and

18 (3) the graded Leibniz rule

hhu|v ∨ wii =hhu|vii∨ w +(−1)|v|(|u|+α)v ∨ hhu|wii. (3.3)

We call hh·|·ii the big bracket on B. Note that by bilinearity and the Leibniz rule, the big bracket vanishes if one of its arguments is in K: hhK|·ii = hh·|Kii = 0. There is another grading on B given by Bk = S k(V[a]∗ ⊕ V[b]). We call this the external degree. There is also a bigrading from

Bk,l = S k,l(V[a]∗ ⊕ V[b]) =∼ S k(V[a]∗) ⊗S l(V[b]) which we call the external bigrading. The big bracket is then of external bidegree (−1, −1) and the symmetric product is of bidegree (0, 0).

Lemma 4. The big bracket satisﬁes the graded Jacobi identity in the following form

hhu|hhv|wiiii = hhhhu|vii|wii +(−1)(|u|+α)(|v|+α)hhv|hhu|wiiii, (3.4) or, equivalently, its symmetric form

0 =(−1)(|u|+α)(|w|+α)hhu|hhv|wiiii +(−1)(|v|+α)(|u|+α)hhv|hhw|uiiii

+(−1)(|w|+α)(|v|+α)hhw|hhu|viiii. (3.5)

Proof. Evaluating the r.h.s. for w = x ∨ y gives

=(−1)(|u|+α)(|w|+α)hhu|hhv|xii∨ yii

+(−1)(|u|+α)(|w|+α)(−1)(|v|+α)|x|hhu|x ∨ hhv|yiiii

+(−1)(|v|+α)(|u|+α)hhv|x ∨ hhy|uiiii

+(−1)(|v|+α)(|u|+α)(−1)|y|(α+|u|)hhv|hhx|uii∨ yii

+(−1)(|w|+α)(|v|+α)x ∨ hhy|hhu|viiii

+(−1)(|w|+α)(|v|+α)(−1)|y|(α+|hhu|vii|)hhx|hhu|viiii∨ y

=(−1)(|u|+α)(|w|+α)hhu|hhv|xiiii∨ y

+(−1)(|u|+α)(|w|+α)(−1)(|u|+α)|hhv|xii|hhv|xii∨ hhu|yii

+(−1)(|u|+α)(|w|+α)(−1)(|v|+α)|x|hhu|xii∨ hhv|yii

+(−1)(|u|+α)(|w|+α)(−1)(|v|+α)|x|(−1)(|u|+α)|x|x ∨ hhu|hhv|yiiii

19 +(−1)(|v|+α)(|u|+α)hhv|xii∨ hhy|uii

+(−1)(|v|+α)(|u|+α)(−1)(|v|+α)|x|x ∨ hhv|hhy|uiiii

+(−1)(|v|+α)(|u|+α)(−1)|y|(α+|u|)hhv|hhx|uiiii∨ y

+(−1)(|v|+α)(|u|+α)(−1)|y|(α+|u|)(−1)(|v|+α)|hhx|uii|hhx|uii∨ hhv|yii

+(−1)(|w|+α)(|v|+α)x ∨ hhy|hhu|viiii

+(−1)(|w|+α)(|v|+α)(−1)|y|(α+|hhu|vii|)hhx|hhu|viiii∨ y

Using |w| = |x| + |y| and the degree of the big bracket, e.g. |hhv|xii| = |v| + |x| + α, we see that of the 10 terms above, terms 2, 5 cancel by anti-symmetry aswell as terms 3, 8. Terms 1, 7, 10 resp. 4, 6, 9 vanish provided that the Jacobi identity holds for the triples (u, v, x) resp. (u, v, y). This proves the Lemma by induction, since we can now reduce the Jacobi identity to the case when u, v, w ∈ V∗[k] ⊕ V[l] are of external degree 1, which is trivial since the big bracket vanishes on K.

We have just seen that for α = 0, B is a graded Poisson algebra, while for α = −1, B is a Gerstenhaber algebra. Sometimes the above structure is called a Poisson (−α)-algebra. Reducing the grading to a Z2-grading by considering only even/odd elements, for even α, B is a Poisson superalgebra, while for odd α, B is a Gerstenhaber superalgebra.

Deﬁnition 15. An L∞ algebra (L, l) consists of

< (1) a Z 0-graded vector space L, and

(2) an element l ∈S(L∗) ⊗ L[α] of degree 1 − α, s.t. hhl|lii = 0.

Lemma 5. Deﬁnitions 15 and 12 are equivalent.

∗ Proof. For an element l ∈ S(L ) ⊗ L[α], we deﬁne dCE(·) = hhl|·ii. This is a derivation of S(L∗) by the Leibniz rule (3.3). Since |l| = 1 − α and the degree of the bracket is α, dCE is of degree +1. Using the Jacobi identity (3.4), we see

2 dCE(ξ)= hhl|hhl|ξiiii (|l|+α)(|l|+α) 2 = hhhhl|lii|ξii +(−1) hhl|hhl|ξiiii = −dCE(ξ),

20 2 since hhl|lii = 0; thus dCE = 0. ∗ Conversely, let dCE be a differential of degree +1 on S(L ). We then deﬁne (implicit summation!)

i |ei|(|e |+α) i l = −(−1) dCE(e ) ∨ ei[α]

i ∗ in terms of a homogeneous basis {ei} of L and its dual basis {e } of L . Since i the degrees of e and ei cancel out, l is of degree 1 − α. With this l,

i j |ei|(|e |+α) i j hhl|e ii = hh−(−1) dCE(e ) ∨ ei[α]|e ii i j = dCE(e ) ∨ hhe |ei[α]ii j = dCE(e ), i.e. dCE(·)= hhl|·ii. Then

j hhl|lii = ±hhl|dCE(e ) ∨ ej[α]ii j j = ±hhl|dCE(e )ii∨ ej[α] ± dCE(e ) ∨ hhl|ej[α]ii,

2 the ﬁrst term vanishes since dCE = 0,

j i j i = ±dCE(e ) ∨ dCE(e ) ∨ hhei[α]|ej[α]ii ±dCE(e ) ∨ hhdCE(e )|ej[α]ii∨ ei[α]

=0 i j i j = ±hhdCE(e )|dCE(e ) ∨ e|j[α]ii∨{z ei[α}] ± hhdCE(e )|dCE(e )ii ∨ej[α] ∨ ei[α] =0 i = ±hhl|dCE(e )ii∨ ei[α]= 0 | {z }

2 again by dCE = 0.

Lemma 6. Deﬁnitions 15 and 9 are equivalent.

Proof. We have already established this equivalence but will nevertheless spell it out explicitly. k ∗ Let (L, l) by an L∞ algebra. We decompose l = ∑ lk, where lk ∈ S (L ) ⊗ L[α] and deﬁne

λk(x1 ∨ ... ∨ xk)= hh... hhlk|x1[α]ii ...|xk[α]ii[−α].

λm(xσ(1) ∨ ... ∨ xσ(m−1) ∨ λn(xσ(m) ∨ ... ∨ xσ(m+n−1))) m m m+n−1 ∑ |x | (1+∑ |xτi|)(1+∑ |x |) − ∑ ǫ(τ)(−1) i=1 τ(i) (−1) i=1 i=m+1 τ(i) −1 τ∈Shm,n−1

λn(xτ(m+1) ∨ ... ∨ xτ(m+n−1) ∨ λm(xτ(1) ∨ ... ∨ xτ(m))) ′ = ∑ ǫ(σ )λm(λn(xσ′(1) ∨ ... ∨ xσ′(n)) ∨ xσ′(n+1) ∨ ... ∨ xσ′(m+n−1)) ′ −1 σ ∈Shn,m−1

+ ∑ ǫ(τ)λn(λm(xτ(1) ∨ ... ∨ xτ(m)) ∨ xτ(m+1) ∨ ... ∨ xτ(m+n−1)) −1 τ∈Shm,n−1

From the above calculation one sees that ∑k+1=m+nhhlm|lnii = 0 is equivalent to the Jacobi identity (3.2). The inverse construction is given by

1 dim L l = ∑ λ (e ∨ ... ∨ e )[α] ∨ eik ∨ ... ∨ ei1 . k k! k i1 ik i1,...,ik=1

We summarize the deﬁnitions and their equivalences in the following diagram.

22 Def. 9: (L, λ) Extension by Def. 11: (S(L), d) λ ∈ Hom1(S(L), L) coderivation d ∈ Coder1(S(L)) Lemma 3 Jacobi (3.2) d2 = 0

λ = projL ◦ d

λk(x1 ∨ ... ∨ xk)= hh· · · hhlk|x1[α]ii· · ·|xk[α]ii[−α] Duality Lemma 6 1 ∑ ik i1 lk = k! λk(ei1 ∨ ... ∨ eik )[α] ∨ e ∨ ... ∨ e ∗ Def. 15: (L, l) Def. 12: (S(L ), dCE) ∗ 1 ∗ l ∈S(L ⊕ L[α])[1 − α] dCE(·)= hhl|·ii dCE ∈ Der (S(L )) Lemma 5 2 hhl|lii = 0 dCE = 0

23 Chapter 4

L∞[α] bialgebras

4.1 L∞[α] bialgebras via the big bracket

Generalizing the discussion in (Kra07) by allowing a shift of degrees, we deﬁne

L∞[α] bialgebras via the big bracket.

Deﬁnition 16. An L∞[α] bialgebra (L, t) consists of

< (1) a Z 0-graded vector space L, and

(2) a degree 1 − α element t = ∑k,l≥1 tkl, where

kl k,l ∗ ∼ k ∗ l tkl ∈ B = S (L ⊕ L[α]) = S (L ) ⊗S (L[α]), such that hht|tii = 0.

The condition that hht|tii = 0 splits into

∑ hhtkl|tk′l′ ii = 0 ∀i, j (4.1) k+k′=i l+l′=j

This implies for j = 2, hh∑ tk1|∑ tk1ii = 0, and thus tk1 deﬁne the structure of an

L∞ algebra on L. On the other hand, the elements

∗ k ∼ ′ k ′∗ t1k ∈ L ⊗S (L[α]) = L [α] ⊗S (L ),

′ ∗ ′ for L = L[α] , also satisfy hh∑ t1k|∑ t1kii = 0. However in general, L need not < be Z 0-graded. This motivates the following proposition.

24 Proposition 7. Let L = L−k ⊕ ... ⊕ L−1 be a k-term graded vector space and (L, t) an L∞[−(k + 1)] bialgebra. Then

(1) (L, l = ∑ tk1) is an L∞ algebra, and

′ ∗ ′ (2) (L = L[α] , l = ∑ t1k) is an L∞ algebra.

We will call such a k-term L∞[−(k + 1)] bialgebra a Lie k-bialgebra.

Proof. Since L is concentrated in degrees −k,..., −1, L[−(k + 1)] is in degrees 1,..., k. Then L[−(k + 1)]∗ is again in degrees −k,..., −1.

′ ′ ∗ <0 We will call (L, l ) an L∞[α] coalgebra, even if L = L[α] is not Z -graded ′ ′ and therefore (L , l ) is not an L∞ algebra. In any case, note that t11 is part of both of the above structures, and that in general there will be elements tkl for k, l ≥ 2. These play the role of homotopies, thus the compatibility between the

L∞ algebra and the L∞[α] coalgebra holds up to higher coherent homotopy. An

L∞[α] bialgebra where tkl vanish for k, l ≥ 2 is said to have strict compatibility.

4.2 Manin L∞[α] triples

′ Deﬁnition 17. A Manin L∞[α] triple is a triple (M; L, L ) of L∞ algebras, where M is equipped with a non-degenerate, graded anti-symmetric bilinear form (·|·) of degree −α, such that

′ ′ (1) M = L ⊕ L as a vector space, and L, L are L∞ subalgebras;

(2) L, L′ are isotropic w.r.t. (·|·), i.e. (L|L) = 0 = (L′|L′);

k (3) the maps µk : S (M) → M deﬁning the L∞ structure are invariant in the sense that

|vk+1||vk| (µk(v1 ∨ ... ∨ vk)|vk+1) =(−1) (µk(v1 ∨ ... ∨ vk−1 ∨ vk+1)|vk) . (4.2)

Theorem 8. Finite-dimensional Manin L∞[α] triples are in one-to-one correspondence to ﬁnite-dimensional Lie (−α − 1)-bialgebras.

25 Proof. To any Lie (−α − 1)-bialgebra (L, t), we can deﬁne a Manin L∞[α] triple (M; L, L′). We let L′ = L[α]∗ and M = L ⊕ L[α]∗. Note that L is in degrees α + 1,..., −1, thus L[α] is in 1,..., −α − 1 and L[α]∗ is again in degrees α + 1,..., −1. We deﬁne the bilinear form for elements u = x + y[α]∗ and v = z + w[α]∗ ∈ M = L ⊕ L[α]∗ by

∗ (x + y[α]∗|z + w[α]∗) = y∗(z) − (−1)|x|(|w |+α)w∗(x).

It is non-degenerate and anti-symmetric. Note that (L|L) = 0 = (L′|L′) as required, and that

∗ ∗ ∗ ∗ hhx[α]+ y |z[α]+ w iiL = (x + y[α] |z + w[α] ) .

∗ The L∞[α] bialgebra structure on L is deﬁned by an element t ∈ S(L ⊕ k ∗ ∼ L[α]) of degree 1 − α. We decompose t = ∑ tk, where tk ∈ S (L ⊕ L[α]) = S k(M∗). We now deﬁne a map M∗ → M[α] by

u∗ = x∗ + y[α]∗∗ 7→ u¯[α]= y[α] − x∗.

Then we check for u, v as above:

u∗(v)=(x∗ + y[α]∗∗)(z + w[α]∗)

∗ = x∗(z)+(−1)|y|(|w |+α)w∗(y)

= (x[α]∗ − y|z + w[α]∗) = − (u¯|v) .

Since the big bracket for M comes from this dual pairing,

∗ ∗ hhu |v¯[α]iiM = u (v¯) = − (u¯|v¯) =(−1)|u¯||v¯| (v¯|u¯)

(|u¯|)(|v∗|+α) ∗ ∗ = −(−1) v (u¯)= hhu¯[α]|v iiM, and

− (u¯|v¯) = − (y − x[α]∗|w − z[α]∗)

∗ = x∗(w) − (−1)|y|(|z |+α)z∗(y)

∗ ∗ ∗ ∗ = hhx + y[α]|z + w[α]iiL = hhu |v iiL.

26 Let us use the isomorphism M∗ =∼ M[α] given above to deﬁne a map c : S k+1(M∗) →S k,1(M∗ ⊕ M[α]) by

k+1 ∗ ∗ ∗ ∗ c : u1 ∨ ... ∨ uk+1 7→ ∑ u1 ∨ ... ∨ u¯i[α] ∨ ... ∨ uk+1. (4.3) i=1

We now check that c(t) deﬁnes an L∞ structure on M. Note that

∗ ∗ ∗ ∗ hhu1 ∨ ... ∨ uk+1|v1 ∨ ... ∨ vl+1iiL k+1 l+1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ = ∑ ∑ ǫ · u1 ∨ ... ∨ ui ∨ ... ∨ uk+1 ∨ hhui |vj ii∨ v1 ∨ ... ∨ vj ∨ ... ∨ vl+1, i=1 j=1 where ǫ denotes the appropriatec sign for the permutation. Onb the other hand,

k+1 l+1 ∗ ∗ ∗ ∗ hh ∑ u1 ∨ ... ∨ u¯r[α] ∨ ... ∨ uk+1| ∑ v1 ∨ ... ∨ v¯s[α] ∨ ... ∨ vk+1iiM r=1 s=1 k+1 l+1 ∗ [ ∗ = ∑ ∑ ǫ · u1 ∨ ... ∨ u¯i[α] ∨ ... ∨ uk+1 i=1 s=1 ∗ [ ∗ ∨ hhu¯i[α]|v1 ∨ ... ∨ v¯s[α] ∨ ... ∨ vl+1iiM ∨ v¯s[α] k+1 l+1 ∗ [ ∗ + ∑ ∑ ǫ · u¯r[α] ∨ hhu1 ∨ ... ∨ u¯r[α] ∨ ... ∨ uk+1|v¯j[α]iiM r=1 j=1 ∗ [ ∗ ∨ v1 ∨ ... ∨ v¯j[α] ∨ ... ∨ vl+1. k+1 l+1 l+1 ∗ [ ∗ = ∑ ∑ ∑ ǫ · u1 ∨ ... ∨ u¯i[α] ∨ ... ∨ uk+1 i=1 j=1 s=1 s6=j

∗ ∗ ∗ ∗ ∨ hhu¯i[α]|vj iiM ∨ v1 ∨ ... ∨ vj ∨ ... ∨ v¯s[α] ∨ ... ∨ vl+1 k+1 l+1 k+1 ∗ ∗ b ∗ + ∑ ∑ ∑ ǫ · u1 ∨ ... ∨ vi ∨ ... ∨ u¯r[α] ∨ ... ∨ uk+1 i=1 j=1 r=1 r6=i b ∗ ∗ [ ∗ ∨ hhvi |v¯j[α]iiM ∨ v1 ∨ ... ∨ v¯j[α] ∨ ... ∨ vl+1.

∗ ∗ ∗ ∗ Since hhu¯i[α]|vj iiM = hhui |vj iiL = hhui |v¯j[α]iiM, the above expression is

∗ ∗ ∗ ∗ = c(hhu1 ∨ ... ∨ uk+1|v1 ∨ ... ∨ vl+1iiL).

This proves that m = c(t) deﬁnes an L∞ structure on M. There are still two ′ conditions to check to see that (M; L, L ) is a Manin L∞ triple.

′ (1) L and L are L∞ subalgebras. This comes from the fact that by deﬁnition

of an L∞[α] bialgebra, t does not contain any terms tkl for k = 0 or l = 0.

Leaving out this subalgebra condition leads to quasi-L∞[α] bialgebras.

27 (2) The invariance condition. Let us check invariance of the derived bracket ∗ ∗ deﬁned by some element c(u1 ∨ ... ∨ uk+1):

∗ Since u¯σ(k+1)|vk+1 = hhuσ(k+1)|vk+1[α]iiM, this proves invariance. By linearity, c(t) deﬁnes an invariant bracket since it is a linear combination of such terms.

Now the converse, i.e. associating to any Manin L∞[α] triple an L∞[α] bialgebra is straight-forward. By non-degeneracy, we can use the bilinear form (·|·) ′ ∗ α α to identify L = L[α] . Then by isotropy and anti-symmetry, hh↓ u|↓ viiL = •,1 ∗ (u|v). The L∞ algebra structure on M is deﬁned by an element m ∈S (M ⊕ M[α]). By invariance this element is in the image of c, i.e. there exists an el- ∗ ement t ∈ S(M ) such that c(t) = m. Then (L, t) is an L∞[α] bialgebra by the same calculation as above. In fact, it is a Lie (−α − 1)-bialgebra for degree reasons.

Following this theorem, we might call Manin L∞[−(k + 1)] triples Manin k-triples.

4.3 Representations up to homotopy

There are several ways to define representations of L∞ algebras. One is, as an L∞-morphism g → gl(E). Then such a representation is called strict, if the morphism is strict. In (LM95), the definition of non-strict morphism is spelled out for the case that the target L∞ algebra is a DGLA. This is the case for gl(E), so it gives a definition for non-strict representations. They show that these non-strict representations are equivalent to the following.

Deﬁnition 18. Let (g, [·]) be an L∞ algebra. A (g, [·])-module (M, ρ) consists of

28 (1) a graded vector space M, and

k−1 (2) a collection of maps ρk : g ⊗ M → M of degree 2 − k. V We require these maps to satisfy the following condition:

i(j−1) ∑ (−1) ∑ χ(σ)ρj(ρi(xσ(1) ∧ ... ∧ xσ(i)) ∧ ... ∧ xσ(k))= 0. i+j=k+1 −1 σ∈Shi,k−i

Note that x1,..., xk−1 ∈ g, xk ∈ M, so either xσ(i) = xk ∈ M or xσ(k) = xk ∈ M. In the ﬁrst case, we deﬁne

ρj(ρi(xσ(1) ∧ ... ∧ xσ(i)) ∧ xσ(i+1) ∧ ... ∧ xσ(k)) ( +∑i ) ∑k =(−1)j−1(−1) i r=1 |xσ(r)| s=i+1 |xσ(s)|

ρj(xσ(i+1) ∧ ... ∧ xσ(k) ∧ ρi(xσ(1) ∧ ... ∧ xσ(i))), where the sign is just the appropriate sign for anti-symmetric permutations. In the second case we take ρi =[·] to be the i-bracket.

As for the deﬁnitions of L∞ algebras, we can give a shifted version of the above deﬁnition.

Deﬁnition 19. Let (L, λ) be an L∞ algebra. A (L, λ)-module (M, ρ) consists of

(1) a graded vector space M, and

k−1 (2) a collection of maps ρk : S (L) ⊗ M → M of degree 1.

We require these maps to satisfy the following condition:

∑ ∑ ǫ(σ)ρj(ρi(xσ(1) ∨ ... ∨ xσ(i)) ∨ xσ(i+1) ∨ ... ∨ xσ(k))= 0. (4.4) i+j=k+1 −1 σ∈Shi,k−i

Note that x1,..., xk−1 ∈ L, xk ∈ M, so either xσ(i) = xk ∈ M or xσ(k) = xk ∈ M. In the ﬁrst case, we deﬁne

ρj(ρi(xσ(1) ∨ ... ∨ xσ(i)) ∨ xσ(i+1) ∨ ... ∨ xσ(k)) ( +∑i ) ∑k =(−1) 1 r=1 |xσ(r)| s=i+1 |xσ(s)|

ρj(xσ(i+1) ∨ ... ∨ xσ(k) ∨ ρi(xσ(1) ∨ ... ∨ xσ(i))), where the sign is just the appropriate sign for symmetric permutations. In the second case we take ρi = λi to be the i-bracket.

29 The proof of equivalence of the above deﬁnitions is very similar to the proof of equivalence of the underlying notions of L∞ algebras, see Lemma 2. From the (co)differential point of view, there is yet another deﬁnition of representation up to homotopy.

∗ Deﬁnition 20. Let (S(L ), dCE) be an L∞ algebra and M a graded vector space. ∗ A representation up to homotopy of (S(L ), dCE) on M is a linear map

∗ ∗ ∗ ∗ DCE : S(L ) ⊗ M →S(L ) ⊗ M

|ξ| 2 of degree +1, s.t. DCE(ξ ∨ η)= dCE(ξ) ∨ η +(−1) ξ ∨ DCE(η) and DCE = 0.

Formally, S(L∗) ⊗ M∗ is a module over the associative algebra S(L∗). Mod- ule multiplication is given by

∇ : S(L∗) ⊗ (S(L∗) ⊗ M∗) →S(L∗) ⊗ M∗, ξ ⊗ (η ⊗ y) 7→ (ξ ∨ η) ⊗ y.

Then DCE is required to be a derivation of this module extending dCE, i.e.

DCE ◦∇ = ∇◦ (dCE ⊗ 1 + 1 ⊗ DCE).

In the dual picture, S(L) ⊗ M is a comodule over the coassociative coalgebra S(L). Comodule comultiplication ∆ : S(L) ⊗ M →S(L) ⊗ (S(L) ⊗ M) is given by

∆ : x1 ∨ ... ∨ xk ⊗ y 7→

∑ ∑ xσ(1) ∨ ... ∨ xσ(i) ⊗ (xσ(i+1) ∨ ... ∨ xσ(k) ⊗ y). i+j=k −1 σ∈Shi,j

Then the following deﬁnition is equivalent to the above by duality.

Deﬁnition 21. Let (S(L), d) be an L∞ algebra and M a graded vector space. A representation up to homotopy of (S(L), d) on M is a coderivation D of the comodule (S(L) ⊗ M, ∆) extending d, i.e.

∆ ◦ D =(d ⊗ 1 + 1 ⊗ D) ◦ ∆.

We require that D2 = 0.

Lemma 9. Deﬁnitions 19 and 21 are equivalent.

30 Proof. The proof is very similar to the proof of Lemma 3.

Let (M, ρ) be a (L, λ)-module as in deﬁnition 19. We decompose λ = ∑ λk k k−1 and ρ = ∑ ρk, where λk : S (L) → L and ρk : S (L) ⊗ M → M. We then extend these by coderivation of the comodule (S(L) ⊗ M, ∆):

Dk(x1 ∨ ... ∨ xn ⊗ y)=

∑ ǫ(σ)λk(xσ(1) ∨ ... ∨ xσ(k)) ∨ xσ(k+1) ∨ ... ∨ xσ(n) ⊗ y −1 σ∈Shk,n−k ∑n−k+1 |x | + ∑ ǫ(σ)(−1) i=1 σ(i) −1 σ∈Shn−k+1,k−1

· xσ(1) ∨ ... ∨ xσ(n−k+1) ∨ ρk(xσ(n−k+2) ∨ ... ∨ xσ(n) ⊗ y) for n + 1 ≥ k. Then D = ∑ Dk. Condition (4.4) is then exactly ρ ◦ D = 0, which is equivalent to D2 = 0 as before.

From D one easily recovers ρ by projection ρ = projM ◦ D.

Now suppose L carries an L∞ structure. We deﬁne the adjoint representation of L on S(L). The adjoint representation on L is simply given by ρk = λk which can then be extended as in the classical case. From the codifferential point of view we can equivalently deﬁne D : S(L) ⊗S(L) → S(L) ⊗S(L) by

D = d ⊗ 1 + 1 ⊗ d, where d is the codifferential describing the L∞ structure on

L. Dually, the adjoint representation is given by DCE = dCE ⊗ 1 + 1 ⊗ dCE.

4.4 L∞ algebra cohomology

Let (S(L), d) be an L∞ algebra, (E, ρ) a representation up to homotopy.

(1) The space of cochains is deﬁned as

• • Cρ = Hom (S(L), E),

k elements of Cρ are the k-cochains.

(2) The coboundary operator δ is deﬁned by

(δc)(x1 ∨ ... ∨ xn)

= ∑ ∑ ǫ(σ)ρk(xσ(1) ∨ ... ∨ xσ(k−1) ∨ c(xσ(k) ∨ ... ∨ xσ(n))) k −1 σ∈Shk−1,n−k+1

+ c(d(x1 ∨ ... ∨ xn)).

31 Note that δ is of degree 1, since ρ and d are of degree 1.

(3) The space of cocyles is the kernel

• • Zρ = {ω ∈ Cρ |δω = 0},

(4) the coboundaries are • •−1 Bρ = δ(Cρ ).

2 2 • • Since δ = 0 by d = 0 and the fact that ρ is a representation, BD ⊆ ZD and we can deﬁne the cohomology • • • HD = ZD/BD.

For the case that L = L−1 is an ordinary Lie algebra and E is ungraded, i.e. in degree zero, the space of cochains is the usual Chevalley-Eilenberg complex for Lie algebras: Homk(S(L), E) = Hom(S k(L), E). Then for degree reasons

ρ = ρ2 is a standard Lie algebra representation and the coboundary operator reduces to the usual one.

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35 Eigenständigkeitserklärung

Hiermit erkläre ich, diese Masterarbeit selbstständig und nur unter Zuhilfe- nahme der angegebenen Quellen verfasst zu haben.

(Malte Dehling)