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 condi- tion 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 struc- ture 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 homo- geneous 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 homoge- neous 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 prod- uct. 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 ho- mogeneous 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.