Cohomology of hemistrict Lie 2-algebras
CCOHOMOLOGY OF HEMISTRICT LIE 2-ALGEBRAS
XIONGWEI CAI, ZHANGJU LIU, AND MAOSONG XIANG
Abstract.
We study representations of hemistrict Lie 2-algebras and give a functorial construction oftheir cohomology. We prove that both the cohomology of an injective hemistrict Lie 2-algebra L and thecohomology of the semistrict Lie 2-algebra obtained from skew-symmetrization of L are isomorphic to theChevalley-Eilenberg cohomology of the induced Lie algebra L Lie . Key words:
Cohomology, Leibniz algebras, Lie 2-algebras.
Contents
1. Introduction 12. Representations of hemistrict Lie 2-algebras 32.1. Dg Leibniz algebras and hemistrict Lie 2-algebras 32.2. Representations of hemistrict Lie 2-algebras 63. Cohomology of hemistrict Lie 2-algebras 93.1. The standard complex 93.2. Cohomology of low orders 113.3. Functoriality 144. Cohomology of injective hemistrict Lie 2-algebras 164.1. Main theorem 174.2. Proof of Theorem 4.4 174.3. Application: Leibniz algebras 21References 221.
Introduction
The notion of weak Lie 2-algebras was introduced by Roytenberg in [18] to complete the picture of cat-egorification of Lie algebras started by Baez and Crans [2]. Roughly speaking, a weak Lie 2-algebra is abilinear bracket on a linear category such that both skew-symmetry and Jacobi identity hold only up tonatural transformations, called alternator and Jacobiator, respectively. It was further shown in op.cit. thatthe 2-category of weak Lie 2-algebras is equivalent to the 2-category of 2-term weak L ∞ -algebras by passingto the normalized chain complex. By a weak L ∞ -algebra, we mean a Loday infinity algebra [1] ( V, { π k } k ≥ )whose structure maps π k are skew-symmetric up to homotopy. Note that the cohomology of the underlyingLoday infinity algebra of a weak L ∞ -algebra defined in loc.cit. by Ammar and Poncin cannot encode theadditional information on the weak symmetry of the structure maps. This is the first of a series of papersdevoted to the study of a new cohomology theory of weak L ∞ -algebras and its relation with other knowncohomology theories. Research partially supported by NSFC grants 11425104, 11901221 and 11931009. a r X i v : . [ m a t h . R A ] M a r XIONGWEI CAI, ZHANGJU LIU, AND MAOSONG XIANG
The purpose of this paper is to study cohomology of hemistrict Lie 2-algebras. Such an algebraic structureis specified by a bilinear bracket [ − , − ] on a 2-term cochain complex (or a 2-vector space) (cid:126)L = L − d −→ L ,which is skew-symmetric up to a chain homotopy h , called alternator, and satisfies the Jacobi (or Leibniz)identity (see Definition 2.9). Thus a hemistrict Lie 2-algebra is indeed a 2-term differential graded (dg forshort) Leibniz algebra whose bracket is skew-symmetric up to homotopy. As an immediate example, eachLeibniz algebra ( g , [ − , − ] g ) gives rise to a hemistrict Lie 2-algebra L g = ( K [1] (cid:44) → g , [ − , − ] g , h ) , where K is the Leibniz kernel, and h is the composition of the degree shifting operator [1] and the K -valuedsymmetric pairing h on g defined by h ( x, y ) = [ x, y ] g + [ y, x ] g for all x, y ∈ g .We first study representations of a hemistrict Lie 2-algebra L = ( (cid:126)L, [ − , − ] , h ) on a 2-term cochain complex (cid:126)V in Section 2. By forgetting the alternator h , it follows that each representation of L gives rise to arepresentation of the dg Leibniz algebra ( (cid:126)L, [ − , − ]). Furthermore, the space of representations of a hemistrictLie 2-algebra L on (cid:126)V is one-to-one correspondent to the space of semidirect products of hemistrict Lie 2-algebras of L by (cid:126)V (see Proposition 2.31).We then focus on a construction of cohomology of hemistrict Lie 2-algebras in Section 3. Our approachoriginates from Roytenberg’s construction of standard complexes for Courant-Dorfman algebras in [19]:Recall that a Courant-Dorfman algebra is quintuple ( R , E , (cid:104)− , −(cid:105) , ∂, [ − , − ]), where R is a commutativealgebra, ( E , (cid:104)− , −(cid:105) ) is a metric R -module, ∂ is an E -valued derivation of R , and [ − , − ] is a Dorfman bracket on E . All the data subject to several compatible conditions generalizing those defining a Courant algebroid [15].Denote by Ω the K¨ahler differential of R . Roytenberg associates to each metric R -module ( E , (cid:104)− , −(cid:105) ) agraded commutative subalgebra C ( E , R ) of the convolution algebra Hom( U ( L ) , R ), where L = E [1] ⊕ Ω [2]is the graded Lie algebra whose bracket is given by the composition d dR (cid:104)− , −(cid:105) of the R -valued metric (cid:104)− , −(cid:105) and the universal de Rham differential d dR : R → Ω . Moreover, the derivation ∂ and the Dorfman bracket[ − , − ] induces a natural differential D on C ( E , R ). The resulting complex was called in op.cit. the standardcomplex of this Courant-Dorfman algebra. Note that the prescribed differential D is indeed defined on thewhole convolution algebra Hom( U ( L ) , R ). This larger cochain complex (Hom( U ( L ) , R ) , D ) includes theinformation of the homotopy term of the underlying hemistrict Lie 2-algebra (Ω [1] → E , {− , −} , d dR (cid:104)− , −(cid:105) )(see Example 3.6), thus to some extent, encodes a new cohomology in need.We associate to each representation V of a hemistrict Lie 2-algebra L = ( (cid:126)L, [ − , − ] , h ) a cochain complex C • ( L, V ), also called the standard complex, where the pair ( (cid:126)L, h ) plays a similar role as the metric R -module in Roytenberg’s construction. The differential is the restriction of the Loday-Pirashvili differential D of the dg Leibniz algebra ( (cid:126)L, [ − , − ]) (see Lemma 3.2). The cohomology of the representation V is definedto be the cohomology of C • ( L, V ) (see Definition 3.4). Applying this construction of standard complex toLeibniz algebras, it is shown in [6] that the Leibniz bracket of a fat Leibniz algebra can be realized as aderived bracket.Note that the construction of standard complexes depends on both a hemistrict Lie 2-algebra and a repre-sentation. It is natural to ask how it varies with respect to these two objects. First of all, when we fix ahemistrict Lie 2-algebra L , the construction of standard complexes is natural with respect to representations(See Proposition 3.18). Meanwhile, the construction of standard complexes is also functorial with respect tothe morphisms of hemistrict Lie 2-algebras (see Theorem 3.23).In Section 4, we study the cohomology of hemistrict Lie 2-algebras of a particular type. A hemistrict Lie2-algebra L = ( (cid:126)L, [ − , − ] , h ) is said to be injective if the differential d of the 2-term cochain complex (cid:126)L isinjective, i.e., H • ( (cid:126)L ) = H ( (cid:126)L ) = L /dL − . The bracket [ − , − ] induces a Lie algebra structure on H ( (cid:126)L ).This Lie algebra is denoted by L Lie . Meanwhile, according to Roytenberg [18], there is a semistrict Lie2-algebra ˜ L obtained from skew-symmetrization. We prove the following Theorem (see Theorem 4.4) . Let L be an injective hemistrict Lie 2-algebra with a representation V suchthat l α = 0 for all α ∈ L − . Then both the Lie algebra L Lie and the semistrict Lie 2-algebra ˜ L admit anatural representation on V . Moreover, H • ( L, V ) ∼ = H • ( ˜ L, V ) ∼ = H • CE ( L Lie , V ) . As an application, let g be a Leibniz algebra and L g the associated hemistrict Lie 2-algebra. Accordingto Roytenberg [18], and independently Sheng and Liu [21], the skew-symmetrization of the Leibniz bracket OHOMOLOGY OF HEMISTRICT LIE 2-ALGEBRAS 3 [ − , − ] g induces a semistrict Lie 2-algebra G = ( K [1] (cid:44) → g , (cid:101) l , (cid:101) l ) . As a consequence, we have
Theorem (see Theorem 4.12) . Let g be a Leibniz algebra with Leibniz kernel K and V a representation ofthe associated hemistrict Lie 2-algebra L g to g such that l α = 0 for all α ∈ K . Then H • ( L g , V ) ∼ = H • ( G , V ) ∼ = H • CE ( g Lie , V ) . The sequel(s).
We plan to write two sequels to this paper, in which we address several issues not coveredhere: In the first one in preparation, we study cohomology of weak Lie 2-algebras, which encodes cohomol-ogy of hemistrict Lie 2-algebras and semistrict Lie 2-algebras into a unified framework. Our first goal is toestablish the compatibility between the functor of taking cohomology and the skew-symmetrization func-tor [18] from the category of weak Lie 2-algebras to the category of semistrict Lie 2-algebras. Meanwhile,in their work [10] of weak Lie 2-bialgebras, Chen, Sti´enon and Xu developed an odd version of big derivedbracket approach to semistrict Lie 2-algebras. Our second goal is to establish a derived bracket formalismfor cohomology of weak Lie 2-algebras.In the second one, we consider weak L ∞ -algebroids which encodes a Courant algebroid as a 2-term weak L ∞ -algebroid. According to Kontsevich and Soibelman [12], L ∞ -algebras correspond to formal pointed dgmanifolds, while A ∞ -algebras correspond to noncommutative formal pointed dg manifolds. Our purposesare to reinterpret weak L ∞ -algebroids as certain commutative up to homotopy formal dg manifolds and toinvestigate their relation with shifted derived Poisson manifolds studied in [4] and strongly homotopy Leibnizalgebra over a commutative dg algebra studied in [9]. Acknowledgement.
We would like to express our gratitude to several institutions for their hospitality whilewe were working on this project: Chern Institute of Mathematics (Xiang), Henan Normal University (Caiand Xiang), Peking University (Cai and Xiang), Tsinghua University (Cai and Xiang). We would also liketo thank Chengming Bai, Zhuo Chen, Yunhe Sheng, Rong Tang, and Tao Zhang for helpful discussions andcomments. Special thanks go to Zhuo Chen for constructive suggestions on this manuscript. We are gratefulto the anonymous referee for carefully reading this paper and providing us valuable suggestions.2.
Representations of hemistrict Lie 2-algebras
In this section, we study representations of hemistrict Lie 2-algebras on 2-term cochain complexes as Beckmodules of the category of hemistrict Lie 2-algebras.2.1.
Dg Leibniz algebras and hemistrict Lie 2-algebras.
As the first step, we collect some basic factsof dg Leibniz algebras, which are Leibniz algebras (or Loday algebras) introduced by Loday [16] in thecategory of cochain complexes (see [11] for more structure theorems on Leibniz algebras). Denote by K thebase field which is either R or C . Definition 2.1.
A dg (left) Leibniz algebra g over K is a (left) Leibniz algebra in the category of cochaincomplexes of K -vector spaces, i.e., a cochain complex (cid:126) g = ( g • , d ) together with a cochain map [ − , − ] g : (cid:126) g ⊗ (cid:126) g → (cid:126) g , called Leibniz bracket, satisfying the (left) Leibniz rule:[ x, [ y, z ] g ] g = [[ x, y ] g , z ] g + ( − | x || y | [ y, [ x, z ] g ] g , ∀ x, y, z ∈ g • . Example 2.2.
Let h = ( h • , d, [ − , − ] h ) be a dg Lie algebra equipped with a representation ρ on a cochaincomplex V = ( V • , d ). Then the direct sum cochain complex h ⊕ V = ( h • ⊕ V • , d ), equipped with the bilinearmap {− , −} defined by { A + v, B + w } (cid:44) [ A, B ] h + ρ ( A ) w, ∀ A, B ∈ h • , v, w ∈ V • , is a dg Leibniz algebra. In particular, the semi-direct product gl ( V ) (cid:110) V for any vector space V , is a Leibnizalgebra, which is called an omni-Lie algebra [23], and is usually denoted by ol ( V ). Furthermore, the sectionspace of an omni-Lie algebroid [8] is also a Leibniz algebra. Remark 2.3.
The notion of 2-term dg Leibniz algebras was investigated by Sheng and Liu in [20] as specialLeibniz 2-algebras. It was shown in loc.cit. that there is a one-to-one correspondence between 2-term dgLeibniz algebras and crossed modules of Leibniz algebras.
XIONGWEI CAI, ZHANGJU LIU, AND MAOSONG XIANG
Definition 2.4.
A left representation V of a dg Leibniz algebra g (or a left g -module) is a cochain complex (cid:126)V = ( V • , d ) equipped with a cochain map l : (cid:126) g → End( (cid:126)V ), called left action, such that l [ x,y ] g = [ l x , l y ] (cid:44) l x ◦ l y − ( − | x || y | l y ◦ l x , for all x, y ∈ g • .A representation of g (or a g -module) is a left representation V = ( (cid:126)V , l ), together with another cochain map r : (cid:126) g → End( (cid:126)V ), called the right action, such that the following conditions hold: r [ x,y ] g = [ l x , r y ] = l x ◦ r y − ( − | x || y | r y ◦ l x , r x ◦ l y = − r x ◦ r y . Given a left g -module V = ( (cid:126)V , l ), there are two standard ways to extend V to a g -module: one is thesymmetric g -module with the right action being r = − l ; the other is the antisymmetric g -module with zeroright action. Example 2.5.
Let g = ( g • , d, [ − , − ] g ) be a dg Leibniz algebra. Then l x y = [ x, y ] g , r x y = ( − | x || y | [ y, x ] g , for all x, y ∈ g • , gives rise to the adjoint representation of g on ( g • , d ). Example 2.6.
Let g be a dg Leibniz algebra. Both the Leibniz kernel K = K ( g ) (cid:44) span { [ x, y ] g + ( − | x || y | [ y, x ] g | x, y ∈ g } and the (left) center Z = Z ( g ) (cid:44) { x ∈ g | [ x, y ] g = 0 , ∀ y ∈ g } are dg ideals of g . The Leibniz bracket induces a left representation l of g on K (or Z ): l x ( y ) (cid:44) [ x, y ] g ∀ x ∈ g , y ∈ K (or Z ) . Moreover, the Leibniz bracket reduces to a Lie bracket on both g /K and g /Z .In [1], Ammar and Poncin defined a cohomology theory for Loday infinity algebras, which generalizes theLoday-Pirashvili cohomology of Leibniz algebras [17]. In particular, we have the following Definition 2.7.
Let g = ( g • , d, [ − , − ] g ) be a dg Leibniz algebra and V = (( V • , d ) , l, r ) a g -module. TheLoday-Pirashvili (bi)complex C • ( g , V ) of the g -module V consists of the following data: • the underlying graded vector space ⊕ n C n ( g , V ) = ⊕ p + q = n Hom(( g • ) ⊗ p , V • ) q , where Hom(( g • ) ⊗ p , V • ) q ⊂ Hom(( g • ) ⊗ p , V • ) consists of elements of degree q ; • the differential D = δ + d V LP : C n ( g , V ) → C n +1 ( g , V ), where(1) δ : Hom(( g • ) ⊗ p , V • ) q → Hom(( g • ) ⊗ p , V • ) q +1 is the internal differential that is specified by thefollowing equation δ ( η )( x , · · · , x p ) = ( − ∗ ( i − p (cid:88) i =1 η ( x , · · · , dx i , · · · , x p ) − ( − n dη ( x , · · · , x p ) , for all η ∈ Hom(( g • ) ⊗ p , V • ) q and x , · · · , x p ∈ g • , where ∗ = 0 and ∗ i = | x | + · · · + | x i | for all1 ≤ i ≤ p .(2) d V LP : Hom(( g • ) ⊗ p , V • ) q → Hom(( g • ) ⊗ ( p +1) , V • ) q , called the Loday-Pirashvili differential, isdefined by( d V LP η )( x , · · · , x p +1 )= p (cid:88) i =1 ( − | x i | ( ∗ i − + n )+ i − l x i η ( x , · · · , (cid:98) x i , · · · , x p +1 ) + ( − | x p +1 | ( n + ∗ p )+ p +1 r x p +1 η ( x , · · · , x p )+ (cid:88) ≤ i The resulting cohomology, denoted by HL • ( g , V ), is called the Loday-Pirashvili cohomology of the g -module V .Next, we recall the 2-category of hemistrict Lie 2-algebras, which is, in fact, a 2-subcategory of 2-term weak L ∞ -algebras [18]: Definition 2.9. A hemistrict Lie 2-algebra L is a 2-term dg Leibniz algebra ( (cid:126)L = ( L − d −→ L ) , [ − , − ]),equipped with a symmetric bilinear map h : L (cid:12) L → L − of degree ( − x, y ] + ( − | x || y | [ y, x ] = d ( h )( x, y ) , (2.10)for all x, y ∈ L • , and [ h ( x, y ) , z ] = 0 , (2.11)[ x, h ( y, z )] = h ([ x, y ] , z ) + h ( y, [ x, z ]) , (2.12)for all x, y, z ∈ L . Here and in the sequel, (cid:12) means the symmetric tensor product over the base field K .A morphism f of hemistrict Lie 2-algebras from L (cid:48) = ( (cid:126)L (cid:48) , [ − , − ] (cid:48) , h (cid:48) ) to L = ( (cid:126)L, [ − , − ] , h ) consists of acochain map f : (cid:126)L (cid:48) → (cid:126)L and a chain homotopy f : L (cid:48) ⊗ L (cid:48) → L − controlling the compatibility between f and the brackets, i.e., [ f ( x (cid:48) ) , f ( y (cid:48) )] − f ([ x (cid:48) , y (cid:48) ] (cid:48) ) = d ( f )( x (cid:48) , y (cid:48) ) , (2.13)for all x (cid:48) , y (cid:48) ∈ L (cid:48)• , such that the following compatible conditions hold: h ( f ( x (cid:48) ) , f ( y (cid:48) )) − f ( h (cid:48) ( x (cid:48) , y (cid:48) )) = f ( x (cid:48) , y (cid:48) ) + f ( y (cid:48) , x (cid:48) ) , (2.14)[ f ( x (cid:48) ) , f ( y (cid:48) , z (cid:48) )] − [ f ( y (cid:48) ) , f ( x (cid:48) , z (cid:48) )] − [ f ( x (cid:48) , y (cid:48) ) , f ( z (cid:48) )] = f ([ x (cid:48) , y (cid:48) ] (cid:48) , z (cid:48) ) + f ( y (cid:48) , [ x (cid:48) , z (cid:48) ] (cid:48) ) − f ( x (cid:48) , [ y (cid:48) , z (cid:48) ] (cid:48) ) , (2.15)for all x (cid:48) , y (cid:48) , z (cid:48) ∈ L (cid:48) . Assume that f (cid:48) is a morphism of hemistrict Lie 2-algebras from L (cid:48)(cid:48) to L (cid:48) . Thecomposition f ◦ f (cid:48) : L (cid:48)(cid:48) → L of f and f (cid:48) is defined by( f ◦ f (cid:48) ) = f ◦ f (cid:48) , ( f ◦ f (cid:48) ) ( x (cid:48)(cid:48) , y (cid:48)(cid:48) ) = f ( f (cid:48) ( x (cid:48)(cid:48) ) , f (cid:48) ( y (cid:48)(cid:48) )) + f ( f (cid:48) ( x (cid:48)(cid:48) , y (cid:48)(cid:48) )) , for all x (cid:48)(cid:48) , y (cid:48)(cid:48) ∈ L (cid:48)(cid:48) .Let f and g be two morphisms of hemistrict Lie 2-algebras from L (cid:48) = ( (cid:126)L (cid:48) , [ − , − ] (cid:48) , h (cid:48) ) to L = ( (cid:126)L, [ − , − ] , h ).A 2-morphism θ : f ⇒ g is a chain homotopy θ : L (cid:48) → L − from f to g , i.e., f ( x (cid:48) ) − g ( x (cid:48) ) = d ( θ )( x (cid:48) ) = dθ ( x (cid:48) ) + θ ( dx (cid:48) ) , ∀ x (cid:48) ∈ L (cid:48)• , satisfying the following condition f ( x (cid:48) , y (cid:48) ) − g ( x (cid:48) , y (cid:48) ) = [ g ( x (cid:48) ) , θ ( y (cid:48) )] + [ θ ( x (cid:48) ) , f ( y (cid:48) )] − θ ([ x (cid:48) , y (cid:48) ] (cid:48) ) = [ f ( x (cid:48) ) , θ ( y (cid:48) )] + [ θ ( x (cid:48) ) , g ( y (cid:48) )] − θ ([ x (cid:48) , y (cid:48) ] (cid:48) ) , for all x (cid:48) , y (cid:48) ∈ L (cid:48) . Remark 2.16. For simplicity, our definition of hemistrict Lie 2-algebra is different from that defined byRoytenberg in [18], where the symmetry assumption on the alternator h is weaker. However, one canapply symmetrization on the alternator of a hemistrict Lie 2-algebra in the sense of Roytenberg to obtain ahemistrict Lie 2-algebra with symmetric alternator as in the above definition.Let L = ( (cid:126)L, [ − , − ] , h ) be a hemistrict Lie 2-algebra. By forgetting the alternator h , it follows immediatelythat ( (cid:126)L, [ − , − ]) is a dg Leibniz algebra. Consequently,(1) ( L , [ − , − ]) is a Leibniz algebra and L − is an L -module.(2) ( H ( (cid:126)L ) , [ − , − ]) is also a Leibniz algebra and H − ( (cid:126)L ) is a symmetric H ( (cid:126)L )-module.According to [18], there associates a natural hemistrict Lie 2-algebra from a Leibniz algebra. Example 2.17. Let ( g , [ − , − ] g ) be a Leibniz algebra with Leibniz kernel K . Let h : g ⊗ g → K be thesymmetric K -valued pairing defined by h ( x, y ) = [ x, y ] g + [ y, x ] g , whose composition with the degree shiftingoperator [1] gives rise to a degree ( − 1) bilinear map h : g ⊗ g h −→ K [1] −→ K [1] . Then L g := ( K [1] (cid:44) → g , [ − , − ] g , h ) is a hemistrict Lie 2-algebra. XIONGWEI CAI, ZHANGJU LIU, AND MAOSONG XIANG Representations of hemistrict Lie 2-algebras. Now we are ready to introduce the notion of rep-resentations of hemistrict Lie 2-algebras: Definition 2.18. Let L = ( (cid:126)L, [ − , − ] , h ) be a hemistrict Lie 2-algebra. A representation of L (or an L -module structure) on a 2-term cochain complex (cid:126)V = ( V − d −→ V ) is specified by the following data: • two cochain maps, called left and right actions, respectively: l : L i ⊗ V j → V i + j , r : V j ⊗ L i → V j + i , (2.19)where i, j = − , V − = 0. • a degree ( − 1) linear map h v : L (cid:12) V → V − , called the action homotopy.The triple ( l, r, h v ) is required to satisfy the following requirements:(1) l gives rise to a left representation of the dg Leibniz algebra ( (cid:126)L, [ − , − ]) on V • , i.e., l x l x v − ( − | x || x | l x l x v = l [ x ,x ] v, (2.20)for all x , x ∈ L • and v ∈ V • , where l x v = l ( x, v ).(2) The actions l, r skew-commute up to homotopy, i.e., l x v + ( − | x || v | r x v = d ( h v )( x, v ) , (2.21)for all x ∈ L • , v ∈ V • , where r x v = ( − | x || v | r ( v, x ).(3) The actions l, r , the action homotopy h v , and the structure maps [ − , − ] , h of L , satisfy the followingcompatible equations: r x h v ( x , v ) = 0 , (2.22) l h ( x ,x ) v = 0 , (2.23) l x h v ( x , v ) = h v ([ x , x ] , v ) + h v ( x , l x v ) , (2.24)for all x , x ∈ L and v ∈ V . Remark 2.25. Since the bracket [ − , − ] and the actions l, r are all cochain maps, it follows that all Equa-tions (2.22)-(2.24) still hold if we replace h v (resp. h ) by d ( h v ) (resp. d ( h )). Example 2.26. Let L = ( (cid:126)L, [ − , − ] , h ) be a hemistrict Lie 2-algebra. There is a natural representation of L on its underlying 2-term cochain complex (cid:126)L with l = r = [ − , − ] and h v = h . This representation, calledthe adjoint representation of L , will be denoted by Ad L . Example 2.27. Let g be a Leibniz algebra and L g the hemistrict Lie 2-algebra as in Example 2.17. Eachordinary representation ( V , l, r ) of the Leibniz algebra g determines a representation V of L g on the 2-termcochain complex (cid:126)V := ( V (cid:48) [1] (cid:44) → V ), where V (cid:48) := { l x v + r x v | x ∈ g , v ∈ V } , the left and right actions areinduced by l, r , and the action homotopy is given by h v ( x, v ) = [1]( l x v ) + [1]( r x v ). In particular, the adjointrepresentation of g on g gives rise to the adjoint representation Ad L g of L g on K [1] → g . Example 2.28. Let L = ( (cid:126)L, [ − , − ] , h ) be a hemistrict Lie 2-algebra. A representation of L on an ordinary(or ungraded) vector space V is, by definition, specified by two linear maps l : L ⊗ V → V, r : V ⊗ L → V, such that ( V, l, r ) is a symmetric representation of the Leibniz algebra ( L , [ − , − ]) and that dL − ⊂ L acts trivially on V . It thus follows that ( V, l, r ) is a symmetric representation of the Leibniz algebra( H ( (cid:126)L ) , [ − , − ]).Analogous to situations in Leibniz algebras, we have some special representations of hemistrict Lie 2-algebras: Definition 2.29. Let L be a hemistrict Lie 2-algebra. A representation V = ( (cid:126)V , l, r, h v ) of L is said to besymmetric if h v = 0, and antisymmetric if r = 0.It follows immediately from Equation (2.21) that any symmetric representation of L is of the form ( (cid:126)V , l, r = − l ) and any antisymmetric representation of L on a 2-term cochain complex (cid:126)V is specified by an “exact”left representation in the following sense: l x v = d ( h v )( x, v ) , ∀ x ∈ L • , v ∈ V • . OHOMOLOGY OF HEMISTRICT LIE 2-ALGEBRAS 7 Example 2.30. Any representation of a hemistrict Lie 2-algebra L on an ordinary vector space is symmetric.Recall that Lie algebra modules are one-to-one correspondent to abelian extensions of the given Lie algebra.For hemistrict Lie 2-algebras, we have the following Proposition 2.31. Let L = ( (cid:126)L, [ − , − ] , h ) be a hemistrict Lie 2-algebra and (cid:126)V := ( V − d −→ V ) a 2-termcomplex. Then the linear maps ( l, r, h v ) as in Definition 2.18 define a representation of L on (cid:126)V if and onlyif the semidirect product L ⊕ V (cid:44) ( −−−−→ L ⊕ V , {− , −} (cid:44) [ − , − ] + l + r, H (cid:44) h + h v )is a hemistrict Lie 2-algebra.To prove this proposition, we need the following result on the relationship between representations of ahemistrict Lie 2-algebra and representations of the associated dg Leibniz algebra. Lemma 2.32. Let V = ( (cid:126)V , l, r, h v ) be a representation of the hemistrict Lie 2-algebra L = ( (cid:126)L, [ − , − ] , h ) .Then ( (cid:126)V , l, r ) is a representation of the dg Leibniz algebra ( (cid:126)L, [ − , − ]) .Proof. Since l gives rise to a left representation of ( (cid:126)L, [ − , − ]) on (cid:126)V , it suffices to check that r [ x,y ] = [ l x , r y ] , r x ◦ l y = − r x ◦ r y , ∀ x, y ∈ L • , or r ( v, [ x, y ]) = l ( x, r ( v, y )) − ( − | x || y | r ( l ( x, v ) , y ) , r x l y v = − ( − | v || y | r x r y v, ∀ v ∈ V • . For the first one, we compute for any v ∈ V − ⊕ V , r ( v, [ x, y ]) = ( − | v | ( | x | + | y | ) r [ x,y ] v by Equation (2.21)= − l [ x,y ] v + d ( h v )([ x, y ] , v ) by Equation (2.20)= − l x l y v + ( − | x || y | l y l x v + d ( h v )([ x, y ] , v ) by Equation (2.21)= ( − | y || v | l x r y v − l x d ( h v )( y, v ) − ( − | v || y | r y l x v + ( − | x || y | d ( h v )( y, l x v ) + d ( h v )([ x, y ] , v ) by Equation (2.24)= ( − | y || v | l x r y v − ( − | x || y | + | y | ( | x | + | v | ) r y l x v = l ( x, r ( v, y )) − ( − | x || y | r ( l ( x, v ) , y )For the second one, by Equation (2.21), we have r x ( r y ( v )) = − ( − | y || v | r x ( l y v ) + r x ( d ( h v )( y, v )) by Equation (2.22)= − ( − | y || v | r x ( l y ( v )) . (cid:3) As an immediate consequence, we have Corollary 2.33. Let V = ( (cid:126)V , l, r, h v ) be a representation of a hemistrict Lie 2-algebra L = ( (cid:126)L, [ − , − ] , h ) .Then both ( V − , l, r ) and ( V , l, r ) are representations of the Leibniz subalgebra ( L , [ − , − ]) of the dg Leibnizalgebra ( (cid:126)L, [ − , − ]) .Proof of Proposition 2.31. Let V = ( (cid:126)V , l, r, h v ) be a representation of L . According to Lemma 2.32, ( (cid:126)V , l, r )is a representation of the dg Leibniz algebra ( (cid:126)L, [ − , − ]). Thus the semidirect product( −−−−→ L ⊕ V , {− , −} = [ − , − ] + l + r )is a dg Leibniz algebra. Meanwhile, it is clear that {− , −} is skewsymmetric up to chain homotopy H = h + h v . Hence, L ⊕ V = ( −−−−→ L ⊕ V , {− , −} = [ − , − ] + l + r, H = h + h v ) is a hemistrict Lie 2-algebra.Conversely, to reconstruct the semidirect product of L by a 2-term cochain complex (cid:126)V , one needs the twocochain maps l, r in Equation (2.19) to recover the underlying semidirect product of dg Leibniz algebraand the chain homotopy h v to control the skewsymmetry of the Leibniz bracket on (cid:126)L ⊕ (cid:126)V . Moreover, it XIONGWEI CAI, ZHANGJU LIU, AND MAOSONG XIANG follows from straightforward verifications that the semidirect product L ⊕ V of hemistrict Lie 2-algebra canbe completed by ( l, r, h v ) only if they give rise to a representation of L on (cid:126)V . (cid:3) Now we study morphisms of representations. Definition 2.34. Let L = ( (cid:126)L, [ − , − ] , h ) be a hemistrict Lie 2-algebra with V = ( (cid:126)V , l, r, h v ), V (cid:48) =( (cid:126)V (cid:48) , l (cid:48) , r (cid:48) , h (cid:48) v ) and V (cid:48)(cid:48) = ( (cid:126)V (cid:48)(cid:48) , l (cid:48)(cid:48) , r (cid:48)(cid:48) , h (cid:48)(cid:48) v ) being its representations.(1) A morphism φ : V → V (cid:48) is a triple ( φ , φ l , φ r ), where φ : (cid:126)V → (cid:126)V (cid:48) is a cochain map, φ l : L ⊗ V → V (cid:48)− and φ r : V ⊗ L → V (cid:48)− are two chain homotopies, such that the following conditions hold: l (cid:48) ( x, φ ( v )) − φ ( l ( x, v )) = d ( φ l )( x, v ) , (2.35) r (cid:48) ( φ ( v ) , x ) − φ ( r ( v, x )) = d ( φ r )( v, x ) ,h (cid:48) v ( x, φ ( v )) − φ ( h v ( x, v )) = φ l ( x, v ) + ( − | x || v | φ r ( v, x ) , for all x ∈ L • and v ∈ V • .(2) Given two morphisms of representations φ = ( φ , φ l , φ r ) : V → V (cid:48) , ψ = ( ψ , ψ l , ψ r ) : V (cid:48) → V (cid:48)(cid:48) , their composition ψ ◦ φ = (( ψ ◦ φ ) , ( ψ ◦ φ ) l , ( ψ ◦ φ ) r ) : V → V (cid:48)(cid:48) is defined by( ψ ◦ φ ) = ψ ◦ φ : (cid:126)V → (cid:126)V (cid:48)(cid:48) , ( ψ ◦ φ ) l = ψ l ◦ (id L ⊗ φ ) + ψ ◦ φ l : L ⊗ V → V (cid:48)(cid:48)− , ( ψ ◦ φ ) r = ψ r ◦ ( φ ⊗ id L ) + ψ ◦ φ r : V ⊗ L → V (cid:48)(cid:48)− . (3) Two representations V, V (cid:48) are said to be isomorphic, if there exist morphisms φ : V → V (cid:48) and ψ : V (cid:48) → V such that ψ ◦ φ = id V := (id (cid:126)V , , 0) and φ ◦ ψ = id V (cid:48) := (id (cid:126)V (cid:48) , , L and their morphisms form a categoryRep( L ), called the category of representations of the hemistrict Lie 2-algebra L . Lemma 2.36. Any object V = ( (cid:126)V , l, r, h v ) in Rep( L ) is isomorphic to the associated symmetric representa-tion V s = ( (cid:126)V , l, − l, .Proof. It follows from straightforward verifications that the pair of morphisms φ = (id (cid:126)V , , − h v ) : V → V s , ψ = (id (cid:126)V , , h v ) : V s → V gives rise to the desired isomorphisms. (cid:3) For this reason, we may view V s as a minimal model of V , which plays a central role in our construction ofcohomology of V in the subsequent section.Finally, we consider pullback representations. Let f = ( f , f ) : L (cid:48) → L be a morphism of hemistrict Lie2-algebras. By Equation (2.13), f does not preserve the brackets [ − , − ] (cid:48) and [ − , − ] strictly but up to achain homotopy f . Thus, representations cannot be pulled back in general. We make the following Definition 2.37. A representation V = ( (cid:126)V , l, r, h v ) of L is said to be f -compatible (or compatible with f ),if the left action l vanishes along the image of the chain homotopy f , i.e., l f ( x (cid:48) ,y (cid:48) ) = 0 ∈ Hom( V , V − ) , for all x (cid:48) , y (cid:48) ∈ L (cid:48) .As an immediate consequence, we have Proposition 2.38. Let f : L (cid:48) → L be a morphism of hemistrict Lie 2-algebras and V an f -compatiblerepresentation. Then there is a pullback representation V (cid:48) of L (cid:48) on (cid:126)V defined by l (cid:48) ( x (cid:48) , v ) = l ( f ( x (cid:48) ) , v ) , r (cid:48) ( v, x (cid:48) ) = r ( v, f ( x (cid:48) )) , h (cid:48) v ( x (cid:48) , v ) = h v ( f ( x (cid:48) ) , v ) , for all x (cid:48) ∈ L (cid:48) and v ∈ V • . Furthermore, if V is symmetric, so is V (cid:48) . Example 2.39. Let f = ( f , f ) : L (cid:48) → L be a morphism of hemistrict Lie 2-algebras. Any representationof L on an ordinary vector space V can be pulled back by f to obtain a pullback representation of L (cid:48) . OHOMOLOGY OF HEMISTRICT LIE 2-ALGEBRAS 9 Cohomology of hemistrict Lie 2-algebras In this section, inspired by the functorial construction of standard cohomology of Courant-Dorfman algebrasby Roytenberg [19], we associate a cochain complex C • ( L, V ), also called standard complex, to each repre-sentation V of a hemistrict Lie 2-algebra L . We prove that this assignment gives rise to a functor from thecategory Rep( L ) of representations of L to the category of cochain complexes on the one hand, and on theother hand a functor from the category of hemistrict Lie 2-algebras to the category of cochain complexes.3.1. The standard complex. Let L = ( (cid:126)L, [ − , − ] , h ) be a hemistrict Lie 2-algebra. The symmetric pairing h : L ⊗ L → L − induces a graded skewsymmetric bilinear map h : L [1] ∧ L [1] → L − [1]. It determinesa graded Lie algebra structure on the vector space L [1]. Let U ( L [1]) = ⊕ n ≥ U ( L [1]) − n = ⊕ n ≥ ⊕ (cid:98) n (cid:99) k =0 (cid:16) ⊗ ( n − k ) ( L [1]) ⊗ S k ( L − [1]) (cid:17) /R be the universal enveloping algebra of L [1], where R is the subspace generated by elements of the form x ⊗ · · · ⊗ x i ⊗ x i +1 ⊗ · · · ⊗ x n − k ⊗ α (cid:12) · · · (cid:12) α k + x ⊗ · · · ⊗ x i +1 ⊗ x i ⊗ · · · ⊗ x n − k ⊗ α (cid:12) · · · (cid:12) α k + x ⊗ · · · ⊗ (cid:98) x i ⊗ (cid:100) x i +1 ⊗ · · · ⊗ x n − k ⊗ h ( x i , x i +1 ) (cid:12) α (cid:12) · · · (cid:12) α k , for all x , · · · , x n − k ∈ L [1] , α , · · · , α k ∈ L − [1] and all 1 ≤ i ≤ n − k − , ≤ k ≤ (cid:98) p (cid:99) .Let V = ( (cid:126)V , l, r, h v ) be a representation of L . Define C • ( L, V ) = ⊕ n ≥− C n ( L, V ) = ⊕ n ≥− ( C n +1 ( L, V − ) ⊕ C n ( L, V ))= ⊕ n ≥− (Hom( U ( L [1]) − ( n +1) , V − ) ⊕ Hom( U ( L [1]) − n , V )) , where our convention is that C − ( L, V ) = 0 and U ( L [1]) − = 0. It follows that the degree n component C n ( L, V ) consists of ( n + 2)-tuples ω = ( ω ( p )0 , ω ( p )1 , · · · , ω ( p ) (cid:98) n − p (cid:99) ) , where p = − ω ( p ) k : ⊗ ( n − p − k ) ( L [1]) ⊗ S k ( L − [1]) → V p , satisfies the following weak symmetry properties: ω ( p ) k ( · · · , x i , x i +1 , · · · | · · · ) + ω ( p ) k ( · · · , x i +1 , x i , · · · | · · · )= − ω ( p ) k +1 ( · · · , (cid:98) x i , (cid:100) x i +1 , · · · | h ( x i , x i +1 ) , · · · ) , (3.1)for all 1 ≤ i ≤ n − p − k − C • ( L, V ). According to Lemma 2.36, the representation V is isomorphicto its minimal model V s = ( (cid:126)V , l, − l, V s . By Lemma 2.32, ( (cid:126)V , l, − l ) is a symmetric representation of the dgLeibniz algebra ( (cid:126)L, [ − , − ]). Moreover, we have the following Lemma 3.2. The differential D of the Loday-Pirashvili complex of the symmetric representation ( (cid:126)V , l, − l ) of ( (cid:126)L, [ − , − ]) determines a differential on C • ( L, V ) D = δ + d LP : C n ( L, V ) → C n +1 ( L, V ) , (3.3) defined for all ω = ( ω ( p )0 , · · · , ω ( p ) (cid:98) n − p (cid:99) ) ∈ C n ( L ; V ) , ( δω ) ( − k ( x , · · · , x n +2 − k | α , · · · , α k ) = k (cid:88) i =1 ω ( − k − ( dα i , x , · · · , x n +2 − k | α , · · · , (cid:98) α i , · · · , α k )( δω ) (0) k ( x , · · · , x n +1 − k | α , · · · , α k ) = k (cid:88) i =1 ω (0) k − ( dα i , x , · · · , x n +1 − k | α , · · · , (cid:98) α i , · · · , α k )+ ( − n +1 d ( ω ( − k ( x , · · · , x n +1 − k | α , · · · , α k )) , ( d LP ω ) (0) k ( x , · · · , x n +1 − k | α , · · · , α k ) = ( d V LP ω (0) k ( · · · | α , · · · , α k ))( x , · · · , x n +1 − k ) + n +1 − k (cid:88) i =1 k (cid:88) j =1 ( − i +1 ω (0) k ( · · · , (cid:98) x i , · · · | · · · , [ α j , x i ] , · · · ) , ( d LP ω ) ( − k ( x , · · · , x n +2 − k | α , · · · , α k ) = d V − LP ( ω ( − k ( · · · | α , · · · , α k ))( x , · · · , x n +2 − k )+ ( − n k (cid:88) j =1 l α j ω (0) k − ( x , · · · , x n +2 − k | α , · · · , (cid:99) α j , · · · , α k )+ n +2 − k (cid:88) i =1 k (cid:88) j =1 ( − i +1 ω ( − k ( · · · , (cid:98) x i , · · · | · · · , [ α j , x i ] , · · · ) , where we have viewed ω ( p ) k ( · · · | α , · · · , α k ) as a degree ( n − p − k ) element of Loday-Pirashvili cochaincomplex of the symmetric L -module ( V p , l, − l ) .Proof. It suffices to show that the differential D preserves the weak symmetry property (3.1). In fact, forany ω ∈ C n ( L, V ),( δω ) ( − k ( · · · , x s , x s +1 , · · · | α , · · · , α k ) + ( δω ) ( − k ( · · · , x s +1 , x s , · · · | α , · · · , α k )= k (cid:88) i =1 ω ( − k − ( dα i , · · · , x s , x s +1 , · · · | · · · , (cid:98) α i , · · · ) + ω ( − k − ( dα i , · · · , x s +1 , x s , · · · | · · · , (cid:98) α i , · · · )= − k (cid:88) i =1 ω ( − k ( dα i , · · · , (cid:98) x s , (cid:91) x s +1 , · · · | h ( x s , x s +1 ) , · · · , (cid:98) α i , · · · )= − ( δω ) ( − k +1 ( · · · , (cid:98) x s , (cid:91) x s +1 , · · · | h ( x s , x s +1 ) , · · · ) + ω ( − k ( dh ( x s , x s +1 ) , · · · , (cid:98) x s , (cid:91) x s +1 , · · · | · · · ) . Meanwhile, since [ h ( x s , x s +1 ) , x i ] = 0 and l ( h ( x s , x s +1 ) , v ) = 0 for all x i ∈ L , v ∈ V , it follows that( d LP ω ) ( − k ( · · · , x s , x s +1 , · · · | · · · ) + ( d LP ω ) ( − k ( · · · , x s +1 , x s , · · · | · · · )= − ( d LP ω ) ( − k +1 ( · · · , (cid:98) x s , (cid:91) x s +1 , · · · | h ( x s , x s +1 ) , · · · ) + ( − s ω ( − k ( · · · , dh ( x s , x s +1 ) , · · · | · · · )+ (cid:88) i By some similar computations (see [5]), one can easily show that ( Dω ) (0) k also satisfies the weak symmetryproperty (3.1). This completes the proof. (cid:3) Definition 3.4. Let L be a hemistrict Lie 2-algebra and V = ( (cid:126)V , l, r, h v ) a representation of L . Wecall ( C • ( L, V ) , D ) the standard complex of the hemistrict Lie 2-algebra L valued in V , whose cohomology H • ( L, V ) is called the cohomology of the representation V of L . Remark 3.5. When the alternator h vanishes, L becomes a strict Lie 2-algebra. In this case, the coho-mology H • ( L, V ) defined above is isomorphic to the generalized Chevalley-Eilenberg cohomology [3, 14] ofthe representation V s = ( (cid:126)V , l, − l ) of the strict Lie 2-algebra L . Example 3.6. Let ( R , E , (cid:104)− , −(cid:105) , ∂, [ − , − ]) be a Courant-Dorfman algebra, where R is a commutative al-gebra, E is an R -module, (cid:104)− , −(cid:105) is an R -valued symmetric R -bilinear form, ∂ is an E -valued derivation of R , and [ − , − ] is a Dorfman bracket on E . All the data are subjected to several conditions (see [19]). Let d dR : R → Ω be the K¨ahler differential of the algebra R . By the universality of Ω , there is a R -modulemorphism ρ ∗ : Ω → E such that ∂ = ρ ∗ ◦ d dR , which is called the coanchor map of E . There exists ahemistrict Lie 2-algebra structure ( {− , −} , h ) on the 2-term cochain complex L ( E ) = ( ρ ∗ : Ω [1] → E )defined as follows: { e , e } (cid:44) [ e , e ] , { α, e } (cid:44) − ι ρ ( e ) d dR α, { e, α } (cid:44) L ρ ( e ) α, h ( e , e ) (cid:44) d dR (cid:104) e , e (cid:105) , for all e , e ∈ E , α ∈ Ω [1], where ρ : E → Der( R ) is the anchor map defined by ρ ( e ) f = (cid:104) e, ∂f (cid:105) for all e ∈ E and f ∈ R . The anchor map ρ determines a symmetric representation of the hemistrict Lie 2-algebra L ( E ).It follows that the standard complex C • ( L ( E ) , R ) of the hemistrict Lie 2-algebra L ( E ) as in Definition 3.4coincides with the convolution dg algebra in [19].3.2. Cohomology of low orders. Let L = ( (cid:126)L, [ − , − ] , h ) be a hemistrict Lie 2-algebra. Thus, ( (cid:126)L, [ − , − ])is a dg Leibniz algebra and L is a Leibniz subalgebra. Let V = ( (cid:126)V , l, r, h v ) be a representation of L . Weconsider the cohomology ⊕ n ≥− H n ( L, V ) of some lower orders.For n = − H − ( L, V ) equals the zeroth Loday-Pirashvili cohomology HL ( L , H − ( (cid:126)V )) of the L -module H − ( (cid:126)V ) = { u ∈ V − | du = 0 } , i.e., H − ( L, V ) = HL ( L , H − ( (cid:126)V )) = { u ∈ V − | du = 0 , l ( x, u ) = r ( u, x ) = 0 , ∀ x ∈ L } . For n = 0, a zeroth-cocycle is a pair ( v, f ), where v ∈ V and f ∈ Hom( L , V − ), satisfying l ( x, v ) = df ( x ) , l ( α, v ) = − f ( dα ) , d V − LP ( f ) = 0 . for all x ∈ L , α ∈ L − . The two equations can be reinterpreted as l ( − , v ) = − D ( f ) : L • → V • . It follows that a zeroth-cocycle is a left ( (cid:126)L, [ − , − ])-invariant element v ∈ V up to a chain homotopy f .Moreover, ( v, f ) is a coboundary if it is of the form ( du, l ( − , u )) for some u ∈ V − .For n = 1, a 1-cocycle ψ is a pair ( ψ , ψ ), where(1) ψ : (cid:126)L → (cid:126)V is a cochain map, i.e., the following diagram commutes: L − d (cid:15) (cid:15) ψ (cid:47) (cid:47) V − d (cid:15) (cid:15) L ψ (cid:47) (cid:47) V ;(2) ψ ∈ HL ( L , V − ), satisfying the weak symmetry (3.1), is the chain homotopy such that ψ : (cid:126)L → (cid:126)V is a derivation of the dg Leibniz algebra ( (cid:126)L, [ − , − ]) up to homotopy, i.e., l ( x, ψ ( y )) + r ( ψ ( x ) , y ) − ψ ([ x, y ]) = dh v ( ψ ( x ) , y ) − dψ ( x, y ) ,l ( α, ψ ( x )) + r ( ψ ( α ) , x ) − ψ ([ α, x ]) = ψ ( dα, x ) − h v ( dψ ( α ) , x ) , for all x, y ∈ L and α ∈ L − . It thus follows that a 1-cocycle ψ is a dg derivation ψ of the dg Leibniz algebra ( (cid:126)L, [ − , − ]) up to homotopyvalued in (cid:126)V . A 1-coboundary, called an inner dg derivation, is of the form( ψ , ψ ) = ( l ( − , v ) − df, d V − LP ( f )) , for some v ∈ V and f ∈ Hom( L , V − ).In particular, let g be a Leibniz algebra and L g = ( K [1] (cid:44) → g , [ − , − ] g , h ) the hemistrict Lie 2-algebra as inExample 2.17. We have Proposition 3.7. There exists a natural injection sending HL ( g , g ) into H ( L g , Ad L g ). Proof. According to Loday and Pirashvili [17], HL ( g , g ) = Der( g , g ) / { inner derivations } . It suffices toassign a(n) (inner) derivation of ( K [1] ⊕ g , [ − , − ] g ) up to homotopy valued in K [1] ⊕ g to each (inner)derivation of g valued in g .On the one hand, each g -valued derivation of g is, by definition, a linear map φ : g → g satisfying φ ([ x, y ] g ) = [ φ ( x ) , y ] g + [ x, φ ( y )] g , ∀ x, y, ∈ g . It follows that φ maps the Leibniz kernel K to itself. Thus, it extends to a cochain map φ : K [1] (cid:44) → g → K [1] (cid:44) → g . Define φ : g ⊗ g → K [1] by φ ( x, y ) = h ( φ ( x ) , y ) = [ φ ( x ) , y ] g + [ y, φ ( x )] g , for all x, y ∈ g . It follows from a straightforward verification that φ = ( φ , φ ) is a 1-cocycle of the adjointrepresentation Ad L g of L g .On the other hand, each inner derivation l x for some x ∈ g gives rise to an inner dg derivation ( r x , 0) ofAd L g . (cid:3) For n = 2, a 2-cocycle ω is a quadruple ω (0)1 : L − → V , ω (0)0 : L ⊗ L → V , ω ( − : L ∧ L − → V − , ω ( − :( L ) ⊗ → V − , (3.8)satisfying the following conditions:(1) ω (0)0 and ω ( − ∈ HL ( L , V − ) satisfy the weak symmetry conditions (3.1).(2) ω (0)1 is ( (cid:126)L, [ − , − ])-invariant up to homotopy, i.e., ω (0)1 ([ α, x ]) − r ( ω (0)1 ( α ) , x ) = dω ( − ( α, x ) − ω (0)0 ( dα, x ) − dh v ( ω (0)1 ( α ) , x ) , (3.9) l ( α, ω (0)1 ( β )) − r ( ω (0)1 ( α ) , β ) = ω ( − ( α, dβ ) − ω ( − ( dα, β ) − h v ( ω (0)1 ( α ) , dβ ) , (3.10)for all x ∈ L and α, β ∈ L − .(3) ω (0)0 and ω ( − are 2-cocycles up to homotopy of the representation of the graded Leibniz algebra( L • , [ − , − ]) on V • , i.e., − dω ( − ( x, y, z ) = d V LP ( ω (0)0 )( x, y, z ) (3.11) h v ( dω ( − ( x | α ) , y ) − ω ( − ( dα, x, y ) = l ( x, ω ( − ( y | α )) + r ( ω ( − ( x | α ) , y ) + l ( α, ω (0)0 ( x, y )) − ω ( − ([ x, y ] | α ) + ω ( − ( y | [ α, x ]) − ω ( − ( x | [ α, y ]) , (3.12)for all x, y ∈ L and α ∈ L − .Given a hemistrict Lie 2-algebra L and an L -module V , recall that an abelian extension of L by V in thecategory of hemistrict Lie 2-algebras (resp. in the category of weak Lie 2-algebras) is a short exact sequenceof hemistrict Lie 2-algebras (resp. weak Lie 2-algebras)0 → V → E → L → , such that the sequence splits as graded vector spaces, the brackets on V is trivial and the action of L on V is the prescribed one. Two such extensions E and E (cid:48) are isomorphic if there exists a morphism of hemistrictLie 2-algebras (resp. weak Lie 2-algebras) from E to E (cid:48) which is compatible with the identity on V and on L . Analogous to abelian extensions of Leibniz algebras [7, 17], we have the following OHOMOLOGY OF HEMISTRICT LIE 2-ALGEBRAS 13 Lemma 3.13. Each second cohomology class [ ω ] ∈ H ( L, V ) of a representation V of the hemistrict Lie2-algebra L gives rise to an equivalence class of abelian extensions of L by V in the category of weak Lie2-algebras.Proof. Let ω = ( ω (0)0 , ω (0)1 , ω ( − , ω ( − ) be a 2-cocycle as in Equation (3.8) of the representation V . Considerthe following binary operation {− , −} defined by { ( x, v ) , ( y, w ) } := ([ x, y ] , l ( x, w ) − l ( y, v ) + ω (0)0 ( x, y )) , { ( x, v ) , ( β, w ) } := ([ x, β ] , l ( x, w ) − l ( β, v ) + ω ( − ( x, β )) , { ( α, v ) , ( y, w ) } := ([ α, y ] , l ( α, w ) − l ( y, v ) + ω ( − ( α, y )) , { ( α, v ) , ( β, w ) } := (0 , l ( α, w ) − l ( β, v )) , for all x, y ∈ L , α, β ∈ L − and v, w ∈ V − ⊕ V , on the cochain complex −−−−→ L ⊕ V = ( L • ⊕ V • , d ω ), wherethe differential d ω is specified by d ω ( α ) = dα − ω (0)1 ( α ) , d ω ( u ) = du, for all α ∈ L − and u ∈ V − . Using Equations (3.9) and (3.10), one can easily verify that {− , −} is indeeda d ω -cochain map.Define a degree ( − 1) map H : ( L • ⊕ V • ) ⊗ → L • ⊕ V • by H (( x, u ) , ( y, v ) , ( z, w )) := (0 , ω ( − ( x, y, z )) . By Equations (3.11) and (3.12), it can also be checked directly that {− , −} satisfies the Jacobi identity upto the chain homotopy H .Finally, define a degree ( − 1) bilinear map H on L • ⊕ V • by H (( x, u ) , ( y, w )) := ( h ( x, y ) , , ∀ x, y ∈ L , u, w ∈ V • . It is clear that { ( x, u ) , ( y, w ) } + { ( y, w ) , ( x, u ) } = d ω ( H (( x, u ) , ( y, w ))) , { ( x, u ) , ( β, w ) } + { ( β, w ) , ( x, u ) } = H (( x, u ) , d ω ( β, w )) , { ( α, u ) , ( y, w ) } + { ( y, w ) , ( α, u ) } = H ( d ω ( α, u ) , ( y, w )) . Hence, L ⊕ V = ( L • ⊕ V • , d ω , {− , −} , H , H ) is a weak Lie 2-algebra. We obtain an extension of L by V → V i =( i ,i ) −−−−−→ L ⊕ V pr=(pr , pr ) −−−−−−−−→ L → , (3.14)where i ( v ) = (0 , v ) , pr ( x, v ) = x for all v ∈ V • and x ∈ L • , and both chain homotopy i and pr vanish.Furthermore, any splitting f = ( f , f ) of short exact sequence (3.14) of weak Lie 2-algebras, if exists, is ofthe form f = id ⊕ ψ : L → L ⊕ V, f = (0 , ψ ) : L ⊗ L → L ⊕ V, where ψ = ( ψ , ψ ) ∈ C ( L, V ) is a 1-cochain. Moreover, it follows from straightforward computations that f is indeed a splitting, i.e., f is a morphism of weak Lie 2-algebras, if and only if D ( ψ ) = ω . This completesthe proof. (cid:3) Remark 3.15. When the alternator h of L vanishes, L becomes a strict Lie 2-algebra and the alternatoron L ⊕ V also vanishes. As a consequence, we rediscover the cohomological description of abelian extensionsof strict Lie 2-algebras in [14].Consider the subset (cid:101) H ( L, V ) ⊂ H ( L, V ) consisting of cohomology classes [ ω ], which have a representative ω = ( ω (0)1 , ω (0)0 , ω ( − , ω ( − ) such that ω ( − = 0. By the argument in the proof of Lemma 3.13, eachelement in (cid:101) H ( L, V ) gives rise to an extension class of L by V in the category of hemistrict Lie 2-algebras.Conversely, it is easy to see that each extension arises in this way. In summary, we have Proposition 3.16. The subset (cid:101) H ( L, V ) is isomorphic to the set of abelian extension classes of L by V inthe category of hemistrict Lie 2-algebras. Remark 3.17. It is natural to consider non-abelian extensions of hemistrict Lie 2-algebras. In [22], Shengand Zhu interpreted non-abelian extensions of Lie algebras as morphisms of Lie 2-algebras. Recently, Liu,Sheng and Wang [13] studied non-abelian extensions of Leibniz algebras by morphisms of Leibniz 2-algebras.Thus, it is expected that non-abelian extensions of hemistrict Lie 2-algebras would be described by morphismsof hemistrict Lie 3-algebras. We will investigate this problem somewhere else.3.3. Functoriality. In this section, we prove that the construction of standard complexes of hemistrict Lie2-algebras is functorial. First of all, we fix a hemistrict Lie 2-algebra L = ( (cid:126)L, [ − , − ] , h ) and prove thatthe assignment of standard complexes to representations of L is functorial. More precisely, we prove thefollowing Proposition 3.18. Assume that φ = ( φ , φ l , φ r ) : V → V (cid:48) is a morphism of representations of L . Thenthere associates a cochain map φ ∗ : ⊕ n C n ( L, V ) −→ ⊕ n C n ( L, V (cid:48) ) , defined by( φ ∗ ω ) (0) k ( x , · · · , x n − k | α , · · · , α k ) = φ ( ω (0) k ( x , · · · , x n − k | α , · · · , α k )) , (3.19)( φ ∗ ω ) ( − k ( x , · · · , x n − k +1 | α , · · · , α k ) = φ ( ω ( − k ( x , · · · , x n − k +1 | α , · · · , α k )) − n − k +1 (cid:88) i =1 ( − n + i φ l ( x i , ω (0) k ( x , · · · , (cid:98) x i , · · · , x n − k +1 | α , · · · , α k )) , (3.20)for all ω ∈ C n ( L, V ), and all x , · · · , x n − k +1 ∈ L , α , · · · , α k ∈ L − . Proof. It follows from straightforward verifications that both ( φ ∗ ω ) (0) k and ( φ ∗ ω ) ( − k satisfy the weak sym-metry property (3.1). Thus φ ∗ is well-defined.Now we show that φ ∗ is a cochain map. For each ω ∈ C n ( L, V ), we compute( φ ∗ Dω ) (0) k ( x , · · · , x n − k +1 | α , · · · , α k ) = φ (( Dω ) (0) k ( x , · · · , x n − k +1 | α , · · · , α k ))= k (cid:88) j =1 φ ( ω (0) k − ( dα j , x , · · · , x n − k +1 | · · · , (cid:99) α j , · · · )) + ( − n +1 φ ( dω ( − k ( x , · · · , x n − k +1 | α , · · · , α k ))+ n − k +1 (cid:88) i =1 ( − i − φ ( l ( x i , ω (0) k ( · · · , (cid:98) x i , · · · | · · · ))) + (cid:88) i Next, we prove that the construction of standard complexes is functorial with respect to morphisms ofhemistrict Lie 2-algebras. More precisely, we have the following Proposition 3.21. Let f = ( f , f ) : L (cid:48) → L be a morphism of hemistrict Lie 2-algebras and V = ( (cid:126)V , l, r )an f -compatible representation of L . Denote by V (cid:48) = ( (cid:126)V , l (cid:48) , r (cid:48) ) the pullback representation of L (cid:48) on (cid:126)V . Then f induces a morphism of standard complexes f ∗ : ⊕ n C n ( L, V ) → ⊕ n C n ( L (cid:48) , V (cid:48) ) , defined by for each ω ∈ C n ( L, V ),( f ∗ ω ) ( p ) k ( x (cid:48) , · · · , x (cid:48) n − k − p | α (cid:48) , · · · , α (cid:48) j )= ω ( p ) k ( f ( x (cid:48) ) , · · · , f ( x (cid:48) n − k − p ) | f ( α (cid:48) ) , · · · , f ( α (cid:48) k )) − (cid:98) n − p (cid:99)− k (cid:88) q =1 (cid:88) i The assignment L → C • ( L, K ) , f (cid:55)→ f ∗ is a contravariant functor from the category ofhemistrict Lie 2-algebras to the category of cochain complexes. To prove Proposition 3.21, one needs, on the one hand, to verify that f ∗ is well defined, i.e., for any ω ∈ C n ( L, V ), f ∗ ω satisfies the following weak symmetry property:( f ∗ ω ) ( p ) k ( · · · , x (cid:48) i , x (cid:48) i +1 , · · · | α (cid:48) , · · · , α (cid:48) j ) + ( f ∗ ω ) ( p ) k ( · · · , x (cid:48) i +1 , x (cid:48) i , · · · | α (cid:48) , · · · , α (cid:48) j )= − ( f ∗ ω ) ( p ) k +1 ( · · · , (cid:98) x (cid:48) i , (cid:100) x (cid:48) i +1 , · · · | h (cid:48) ( x (cid:48) i , x (cid:48) i +1 ) , α (cid:48) , · · · , α (cid:48) j ) , (3.24)and on the other hand, to prove that f ∗ is a cochain map, i.e.,( Df ∗ ω ) ( p ) k ( x , · · · , x n − k − p +1 | α , · · · , α k ) = ( f ∗ Dω ) ( p ) k ( x , · · · , x n − k − p +1 | α , · · · , α k ) . (3.25)In fact, both Equation (3.24) and Equation (3.25) follow from a straightforward but tedious calculation byusing the following equations h ( f ( x (cid:48) ) , f ( x (cid:48) )) − f ( h (cid:48) ( x (cid:48) , x (cid:48) )) = f ( x (cid:48) , x (cid:48) ) + f ( x (cid:48) , x (cid:48) ) , (3.26)[ f ( x (cid:48) ) , f ( x (cid:48) )] − f ([ x (cid:48) , x (cid:48) ] (cid:48) ) = d ( f )( x (cid:48) , x (cid:48) ); (3.27)[ f ( x (cid:48) ) , f ( x (cid:48) , x (cid:48) )] − [ f ( x (cid:48) ) , f ( x (cid:48) , x (cid:48) )] − [ f ( x (cid:48) , x (cid:48) ) , f ( x (cid:48) )]= f ([ x (cid:48) , x (cid:48) ] (cid:48) , x (cid:48) ) + f ( x (cid:48) , [ x (cid:48) , x (cid:48) ] (cid:48) ) − f ( x (cid:48) , [ x (cid:48) , x (cid:48) ] (cid:48) ) , (3.28)for all x (cid:48) , x (cid:48) , x (cid:48) ∈ L (cid:48)• , by the definition of morphisms of hemistrict Lie 2-algebras, and l (cid:48) x (cid:48) v = l f ( x (cid:48) ) v, ∀ x (cid:48) ∈ L (cid:48)• , v ∈ V • , (3.29)by the definition of pullback representations. To save space and time, we omit the proof. However, inorder to see how the above equations are involved in calculations, we verify Equation (3.24) in the case that ψ = ( ψ , ψ ) ∈ C ( L, V ) and Equation (3.25) in the case that ω = ( ω (0)0 , ω (0)1 , ω ( − , ω ( − ) ∈ C ( L, V ),respectively, by proving the following( f ∗ ψ ) ( x (cid:48) , x (cid:48) ) + ( f ∗ ψ ) ( x (cid:48) , x (cid:48) ) = − ( f ∗ ψ ) ( h (cid:48) ( x (cid:48) , x (cid:48) ));( f ∗ Dω ) (0)0 ( x (cid:48) , x (cid:48) , x (cid:48) ) = ( Df ∗ ω ) (0)0 ( x (cid:48) , x (cid:48) , x (cid:48) ) . For the first one, we compute( f ∗ ψ ) ( x (cid:48) , x (cid:48) ) + ( f ∗ ψ ) ( x (cid:48) , x (cid:48) ) = ψ ( f ( x (cid:48) ) , f ( x (cid:48) )) + ψ ( f ( x (cid:48) ) , f ( x (cid:48) )) + ψ ( f ( x (cid:48) , x (cid:48) )) + ψ ( f ( x (cid:48) , x (cid:48) ))= − ψ ( h ( f ( x (cid:48) ) , f ( x (cid:48) ))) + ψ ( f ( x (cid:48) , x (cid:48) ) + f ( x (cid:48) , x (cid:48) )) by Equation (3.26)= − ψ ( f ( h (cid:48) ( x (cid:48) , x (cid:48) ))) = − ( f ∗ ψ ) ( h (cid:48) ( x (cid:48) , x (cid:48) )) . For the second equation, we compute, on the one hand,( f ∗ Dω ) (0)0 ( x (cid:48) , x (cid:48) , x (cid:48) ) = ( Dω ) (0)0 ( f ( x (cid:48) ) , f ( x (cid:48) ) , f ( x (cid:48) ))+ ( Dω ) (0)1 ( f ( x (cid:48) ) | f ( x (cid:48) , x (cid:48) )) − ( Dω ) (0)1 ( f ( x (cid:48) ) | f ( x (cid:48) , x (cid:48) )) + ( Dω ) (0)1 ( f ( x (cid:48) ) | f ( x (cid:48) , x (cid:48) )) , where( Dω ) (0)0 ( f ( x (cid:48) ) , f ( x (cid:48) ) , f ( x (cid:48) )) = − dω ( − ( f ( x (cid:48) ) , f ( x (cid:48) ) , f ( x (cid:48) ))+ l f ( x (cid:48) ) ω (0)0 ( f ( x (cid:48) ) , f ( x (cid:48) )) − l f ( x (cid:48) ) ω (0)0 ( f ( x (cid:48) ) , f ( x (cid:48) )) + l f ( x (cid:48) ) ω (0)0 ( f ( x (cid:48) ) , f ( x (cid:48) )) − ω (0)0 ([ f ( x (cid:48) ) , f ( x (cid:48) )] , f ( x (cid:48) )) − ω (0)0 ( f ( x (cid:48) ) , [ f ( x (cid:48) ) , f ( x (cid:48) )]) + ω (0)0 ( f ( x (cid:48) ) , [ f ( x (cid:48) ) , f ( x (cid:48) )]) , and ( Dω ) (0)1 ( f ( x (cid:48) ) | f ( x (cid:48) , x (cid:48) )) = − dω ( − ( f ( x (cid:48) ) | f ( x (cid:48) , x (cid:48) )) + ω (0)0 ( df ( x (cid:48) , x (cid:48) ) , f ( x (cid:48) ))+ l f ( x (cid:48) ) ω (0)1 ( f ( x (cid:48) , x (cid:48) )) + ω (0)1 ([ f ( x (cid:48) , x (cid:48) ) , f ( x (cid:48) )]) , ( Dω ) (0)1 ( f ( x (cid:48) ) | f ( x (cid:48) , x (cid:48) )) = − dω ( − ( f ( x (cid:48) ) | f ( x (cid:48) , x (cid:48) )) + ω (0)0 ( df ( x (cid:48) , x (cid:48) ) , f ( x (cid:48) ))+ l f ( x (cid:48) ) ω (0)1 ( f ( x (cid:48) , x (cid:48) )) + ω (0)1 ([ f ( x (cid:48) , x (cid:48) ) , f ( x (cid:48) )]) , ( Dω ) (0)1 ( f ( x (cid:48) ) | f ( x (cid:48) , x (cid:48) )) = − dω ( − ( f ( x (cid:48) ) | f ( x (cid:48) , x (cid:48) )) + ω (0)0 ( df ( x (cid:48) , x (cid:48) ) , f ( x (cid:48) ))+ l f ( x (cid:48) ) ω (0)1 ( f ( x (cid:48) , x (cid:48) )) + ω (0)1 ([ f ( x (cid:48) , x (cid:48) ) , f ( x (cid:48) )]) . On the other hand, we have( Df ∗ ω ) (0)0 ( x (cid:48) , x (cid:48) , x (cid:48) ) = ( δf ∗ ω ) (0)0 ( x (cid:48) , x (cid:48) , x (cid:48) ) + ( d LP f ∗ ω ) (0)0 ( x (cid:48) , x (cid:48) , x (cid:48) ) , where ( δf ∗ ω ) (0)0 ( x (cid:48) , x (cid:48) , x (cid:48) ) = − dω ( − ( f ( x (cid:48) ) , f ( x (cid:48) ) , f ( x (cid:48) )) − dω ( − ( f ( x (cid:48) ) | f ( x (cid:48) , x (cid:48) ))+ dω ( − ( f ( x (cid:48) ) | f ( x (cid:48) , x (cid:48) )) − dω ( − ( f ( x (cid:48) ) | f ( x (cid:48) , x (cid:48) )) , and( d LP f ∗ ω ) (0)0 ( x (cid:48) , x (cid:48) , x (cid:48) ) = l (cid:48) x (cid:48) ( f ∗ ω ) (0)0 ( x (cid:48) , x (cid:48) ) − l (cid:48) x (cid:48) ( f ∗ ω ) (0)0 ( x (cid:48) , x (cid:48) ) + l (cid:48) x (cid:48) ( f ∗ ω ) (0)0 ( x (cid:48) , x (cid:48) ) − ( f ∗ ω ) (0)0 ([ x (cid:48) , x (cid:48) ] (cid:48) , x (cid:48) ) − ( f ∗ ω ) (0)0 ( x (cid:48) , [ x (cid:48) , x (cid:48) ] (cid:48) ) + ( f ∗ ω ) (0)0 ( x (cid:48) , [ x (cid:48) , x (cid:48) ] (cid:48) )= l (cid:48) x (cid:48) ω (0)0 ( f ( x (cid:48) ) , f ( x (cid:48) )) − l (cid:48) x (cid:48) ω (0)0 ( f ( x (cid:48) ) , f ( x (cid:48) )) + l (cid:48) x (cid:48) ω (0)0 ( f ( x (cid:48) ) , f ( x (cid:48) ))+ l (cid:48) x (cid:48) ω (0)1 ( f ( x (cid:48) , x (cid:48) )) − l (cid:48) x (cid:48) ω (0)1 ( f ( x (cid:48) , x (cid:48) )) + l (cid:48) x (cid:48) ω (0)1 ( f ( x (cid:48) , x (cid:48) )) − ω (0)0 ( f ([ x (cid:48) , x (cid:48) ] (cid:48) ) , f ( x (cid:48) )) − ω (0)0 ( f ( x (cid:48) ) , f ([ x (cid:48) , x (cid:48) ] (cid:48) )) + ω (0)0 ( f ( x (cid:48) ) , f ([ x (cid:48) , x (cid:48) ] (cid:48) )) − ω (0)1 ( f ([ x (cid:48) , x (cid:48) ] (cid:48) , x (cid:48) )) − ω (0)1 ( f ( x (cid:48) , [ x (cid:48) , x (cid:48) ] (cid:48) )) + ω (0)1 ( f ( x (cid:48) , [ x (cid:48) , x (cid:48) ] (cid:48) )) . Then it follows from Equations (3.27), (3.28), (3.29), and a direct verification that( f ∗ Dω ) (0)0 ( x (cid:48) , x (cid:48) , x (cid:48) ) = ( Df ∗ ω ) (0)0 ( x (cid:48) , x (cid:48) , x (cid:48) ) . Cohomology of injective hemistrict Lie 2-algebras A hemistrict Lie 2-algebra L = ( (cid:126)L, [ − , − ] , h ) is said to be injective, if the differential d of the underlying2-term complex (cid:126)L = L − d −→ L is injective, i.e., there is a short exact sequence of graded vector spaces0 → L − d −→ L −→ L /dL − → . (4.1)Let L = ( (cid:126)L, [ − , − ] , h ) be an injective hemistrict Lie 2-algebra. The vector space H • ( (cid:126)L ) = L /dL − , togetherwith the bracket {− , −} defined by { ¯ x, ¯ y } = pr([ x, y ]) , ∀ ¯ x, ¯ y ∈ L /dL − , OHOMOLOGY OF HEMISTRICT LIE 2-ALGEBRAS 17 for all x, y ∈ L such that pr( x ) = ¯ x, pr( y ) = ¯ y , is a Lie algebra, which will be denoted by L Lie . Meanwhile,according to Roytenberg [18], the skew-symmetrization on the bracket [ − , − ] gives rise to a semistrict Lie2-algebra ˜ L = ( (cid:126)L, ˜ l , ˜ l ), where˜ l ( x, y ) = 12 (cid:16) [ x, y ] − ( − | x || y | [ y, x ] (cid:17) , ∀ x, y ∈ L − ⊕ L , (4.2)˜ l ( x, y, z ) = − (cid:16) h (˜ l ( x, y ) , z ) + h (˜ l ( y, z ) , x ) + h (˜ l ( z, x ) , y ) (cid:17) , ∀ x, y, z ∈ L . (4.3)The main purpose of this section is to build isomorphisms of these three objects on the cohomology level.4.1. Main theorem. Assume that V = ( (cid:126)V , l, r, h v ) is a representation of L such that l α = 0 for all α ∈ L − .It follows that l induces a representation of the Lie algebra L Lie as well as a representation of the semistrictLie 2-algebra ˜ L on (cid:126)V , i.e., V is both an L Lie -module and an ˜ L -module.Here is our main theorem: Theorem 4.4. Let L be an injective hemistrict Lie 2-algebra and V = ( (cid:126)V , l, r, h v ) a representation suchthat l α = 0 for all α ∈ L − .(1) Assume that the alternator h satisfies h ( dα, dβ ) = 0 for any α, β ∈ L − . Then the cohomologyof the representation V of L is isomorphic to the Chevalley-Eilenberg cohomology of the Lie algebra L Lie with coefficient V , i.e., H • ( L, V ) ∼ = H • CE ( L Lie , V ) . (2) The cohomology of the semistrict Lie 2-algebra ˜ L obtained from skew-symmetrization is isomorphicto the Chevalley-Eilenberg cohomology of the Lie algebra L Lie , i.e., H • ( ˜ L, V ) ∼ = H • CE ( L Lie , V ) . As a consequence, we have Corollary 4.5. Under the assumptions as in Theorem 4.4, we have H • ( L, V ) ∼ = H • ( ˜ L, V ) . Remark 4.6. In fact, each representation V of a hemistrict Lie 2-algebra L = ( (cid:126)L, [ − , − ] , h ) induces arepresentation { µ k } k =1 of the semistrict Lie 2-algebra ˜ L obtained from L via skew-symmetrization on (cid:126)V ,where µ = d : V − → V , µ = l : L • ⊗ V • → V • , and µ : L ∧ L ⊗ V → V − is specified by µ ( x, y, v ) = 12 l ( h ( x, y ) , v ) , for all x, y ∈ L and v ∈ V . It is natural to ask if the isomorphism in Corollary 4.5 holds for generalrepresentations. We will investigate this problem in a incoming paper.4.2. Proof of Theorem 4.4. Proof of the first statement. Note that the associate Lie algebra L Lie may also be viewed as a hemistrictLie 2-algebra (0 ⊕ L /dL − , [ − , − ] , h = 0). The projection pr : L → L /dL − extends to a morphism ofhemistrict Lie 2-algebras f := (pr , 0) : L −→ L Lie . The first observation is the following Lemma 4.7. Let j : L /dL − → L be a splitting of (4.1) . There associates a morphism of hemistrict Lie2-algebras g := ( g , g ) : L Lie −→ L, defined by g (¯ x ) = j (¯ x ) , g (¯ x, ¯ y ) = pr − ([ j (¯ x ) , j (¯ y )]) , for all ¯ x, ¯ y ∈ L /dL − , where pr − : L → L − is the projection specified by id L = j ◦ pr + d ◦ pr − , such that f ◦ g = id : L Lie → L Lie . Furthermore, the map θ := pr − : L → L − gives rise to a 2-morphism θ : id L ⇒ g ◦ f .Proof. We first show that g is well-defined by verifying Equations (2.13), (2.14) and (2.15) in this case. Infact, we have, by definition,[ g (¯ x ) , g (¯ x )] − g ( { ¯ x, ¯ y } ) = [ j (¯ x ) , j (¯ y )] − j (pr([ j (¯ x ) , j (¯ y )]))= d pr − ([ j (¯ x ) , j (¯ y )]) = dg (¯ x, ¯ y ) ,g (¯ x, ¯ y ) + g (¯ y, ¯ x ) = pr − ([ j (¯ x ) , j (¯ y )] + [ j (¯ y ) , j (¯ x )]) = pr − ( dh ( j (¯ x ) , j (¯ y )))= h ( j (¯ x ) , j (¯ y )) = h ( g (¯ x ) , g (¯ y )) , and [ g (¯ x ) , g (¯ y, ¯ z )] − [ g (¯ y ) , g (¯ x, ¯ z )] − [ g (¯ x, ¯ y ) , ¯ z ] − g ( { ¯ x, ¯ y } , ¯ z ) − g (¯ y, { ¯ x, ¯ z } ) + g (¯ x, { ¯ y, ¯ z } )= pr − ([ j (¯ x ) , [ j (¯ y ) , j (¯ z )]] − [ j (¯ y ) , [ j (¯ x ) , j (¯ z )]] − [[ j (¯ x ) , j (¯ y )] , j (¯ z )]) = 0 . Meanwhile, since( f ◦ g ) = f ◦ g = pr ◦ j = id , ( f ◦ g ) = f ◦ ( g ⊗ g ) + f ◦ g = 0 , it follows that f ◦ g = id : L Lie → L Lie .Finally, since on the one hand( g ◦ f ) ( x ) = g ( f ( x )) = j (pr( x )) = x − d pr − ( x ) = x − d ( θ )( x ) , ∀ x, y ∈ L • , thus θ is a chain homotopy from id L to g ◦ f . On the other hand, note that( g ◦ f ) ( x, y ) = g ( f ( x ) , f ( y )) + g ( f ( x, y )) = pr − ([ j (pr( x )) , j (pr( y ))]) = pr − ([ x − dθ ( x ) , y − dθ ( y )])= θ ([ x, y ]) − pr − ( d [ x, θ ( y )]) − pr − ( d [ θ ( x ) , y ]) + pr − ( d [ dθ ( x ) , θ ( y )])= θ ([ x, y ]) − [ x, θ ( y )] − [ θ ( x ) , y ] + [ dθ ( x ) , θ ( y )]= θ ([ x, y ]) − [( g ◦ f ) ( x ) , θ ( y )] − [ θ ( x ) , y ] , for all x, y ∈ L . Thus, θ : id L ⇒ g ◦ f is a 2-morphism. (cid:3) Let V = ( (cid:126)V , l, r, h v ) be a representation of L such that l α = 0 for all α ∈ L − . By Lemma 4.7, V is( g ◦ f )-compatible, and the pullback representation by g ◦ f coincides with V . By Proposition 3.21, we havea cochain map ( g ◦ f ) ∗ : C • ( L, V ) → C • ( L, V ) . Moreover, we have the following Lemma 4.8. Under the assumption that h ( dα, dβ ) = 0 for any α, β ∈ L − , the 2-morphism θ : id L ⇒ g ◦ f gives rise to a 2-morphism Θ : id C • ( L,V ) ⇒ ( g ◦ f ) ∗ in the 2-category of cochain complexes, i.e., there existsa chain homotopy Θ : C • ( L, V ) → C •− ( L, V ) such that for all ω ∈ C • ( L, V ) , ω − ( g ◦ f ) ∗ ( ω ) = D Θ( ω ) + Θ( Dω ) . Proof. For each ω ∈ C • ( L, V ), define(Θ ω ) ( p ) k ( x , · · · , x n − k − p − | α , · · · , α k )= (cid:98) n − p (cid:99)− k − (cid:88) q =0 (cid:88) i To prove the second statement of Theorem 4.4, we need the homologicalperturbation lemma (cf. [4]), which we recall as follows:Let us start with a homotopy contraction of cochain complexes:( A, d A ) ( B, d B ) , h φψ where both ψ and φ are maps of cochain complexes, and h is the degree ( − 1) chain homotopy, satisfying thefollowing two equations φ ◦ ψ = id B , ψ ◦ φ = id A − [ d A , h ] , together with the side conditions h ◦ ψ = 0 , φ ◦ h = 0 , h = 0 . Lemma 4.9 (The Perturbation Lemma) . Let ( A, D A = d A + ρ ) be a perturbation of ( A, d A ) . Then we havea new homotopy contraction ( A, D A ) ( B, D B ) , H ΦΨ where D B = d B + (cid:88) k ≥ φ ( ρh ) k ρψ, Φ = (cid:88) k ≥ φ ( ρh ) k ,H = (cid:88) k ≥ ( hρ ) k h = (cid:88) k ≥ h ( ρh ) k , Ψ = (cid:88) k ≥ ( hρ ) k ψ. Now we analyze the two cochain complexes in our situation: We will denote by K ⊂ L the image of d : L − → L in the sequel.On the one hand, the Chevalley-Eilenberg cochain complex of the Lie algebra L Lie is B := ⊕ n ≥ C n ( L Lie , V ) = ⊕ n ≥ ⊕ p + r = n ∧ p ( L /K ) ∨ ⊗ V r , with the differential D B = d V + d CE , where(1) d V comes from the differential of the 2-term complex (cid:126)V , which increases the index r by 1, i.e., d V : ∧ p ( L /K ) ∨ ⊗ V r → ∧ p ( L /K ) ∨ ⊗ V r +1 , where V r = 0 if r (cid:54) = − , d CE is the Chevalley-Eilenberg differential of the Lie algebra L Lie with coefficient V • = V − ⊕ V ,which increases the index p by 1, i.e., d CE : ∧ p ( L /K ) ∨ ⊗ V r → ∧ p +1 ( L /K ) ∨ ⊗ V r . On the other hand, the cochain complex of the semistrict Lie 2-algebra ˜ L with coefficient V is, by definition, A := C • ( ˜ L, V ) = ⊕ n ≥ C n ( ˜ L, V ) = ⊕ n ≥ ⊕ p +2 q + r = n ∧ p ( L ) ∨ ⊗ S q (( L − [1]) ∨ ) ⊗ V r , with the differential D A = d V + δ + ρ , where(1) d V also comes from the differential of the 2-term complex (cid:126)V , which increases the index r by 1, i.e., d V : ∧ p ( L ) ∨ ⊗ S q (( L − [1]) ∨ ) ⊗ V r → ∧ p ( L ) ∨ ⊗ S q (( L − [1]) ∨ ) ⊗ V r +1 ; (2) δ is induced from the differential of the 2-term complex (cid:126)L , which decreases the index p by 1 andincreases the index q by 1 at the same time, i.e., δ : (cid:77) p,q ≥ ∧ p +1 ( L ) ∨ ⊗ S q (( L − [1]) ∨ ) ⊗ V r → (cid:77) p,q ≥ ∧ p ( L ) ∨ ⊗ S q +1 (( L − [1]) ∨ ) ⊗ V r ;(3) ρ = − ˜ l ∨ + ˜ l ∨ + l ∨ is the sum of duals of the 2-bracket ˜ l , the 3-bracket ˜ l , defined by Equa-tion (4.2), (4.3), and the left action l of L on V • .It is clear that ( d V + δ ) = 0. Thus, the cochain complex A results from a perturbation of the complex A (cid:48) := ( ⊕ n ≥ C n ( ˜ L, V ) , d V + δ ) . Let us choose a splitting j : L /K → L of the following exact sequence of vector spaces0 → K i −→ L −→ L /K → . Thus, L ∼ = K ⊕ L /K and ∧ p ( L ) ∨ ∼ = ⊕ t + s = p ∧ t ( L /K ) ∨ ⊗ ∧ s K ∨ .The first observation is the following Lemma 4.10. There is a homotopy contraction A (cid:48) := ( ⊕ n ≥ C n ( ˜ L, V ) , d V + δ ) ( ⊕ n ≥ ⊕ p + r = n ∧ p ( L /K ) ∨ ⊗ V r , d V ) := B (cid:48) , h φψ where ψ : ⊕ p + r = n ∧ p ( L /K ) ∨ ⊗ V r pr ∨ ⊗ id V r −−−−−−−→ ⊕ p + r = n ∧ p ( L ) ∨ ⊗ V r (cid:44) → C n ( ˜ L, V ) ,φ : C n ( ˜ L, V ) (cid:16) ⊕ p + r = n ∧ p ( L ) ∨ ⊗ V r j ∨ ⊗ id V r −−−−−−→ ⊕ p + r = n ∧ p ( L /K ) ∨ ⊗ V r , and the degree ( − chain homotopy h : ∧ t ( L /K ) ∨ ⊗ ∧ s K ∨ ⊗ S q +1 (( L − [1]) ∨ ) ⊗ V r → ∧ t ( L /K ) ∨ ⊗ ∧ s +1 K ∨ ⊗ S q (( L − [1]) ∨ ) ⊗ V r is specified by for all ω ∈ Hom( ∧ t L /K ⊗ ∧ s K ⊗ S q +1 ( L − [1]) , V r ) ∼ = ∧ t ( L /K ) ∨ ⊗ ∧ s K ∨ ⊗ S q +1 (( L − [1]) ∨ ) ⊗ V r ,h ( ω )(¯ x , · · · , ¯ x t , k , · · · , k s +1 | α , · · · , α q )= (cid:40) s + q (cid:80) s +1 j =1 ( − s +1 − j ω (¯ x , · · · , ¯ x t , · · · , (cid:98) k j , · · · | pr − ( k j ) , α , · · · , α q ) , if s + q > , , otherwise , for all ¯ x , · · · , ¯ x t ∈ L /K, k , · · · , k s +1 ∈ K, α , · · · , α q ∈ L − . Here pr − : K → L − is the inverse of theisomorphism d : L − → K ⊂ L . The proof of this lemma is straightforward, thus omitted.Applying the perturbation Lemma 4.9 to the contraction in Lemma 4.10, we prove the following Lemma 4.11. There is a homotopy contraction A = ( ⊕ n ≥ C n ( ˜ L, V ) , D A = d V + δ + ρ ) B = ( ⊕ n ≥ ∧ n ( L /K ) ∨ ⊗ V, D B := d V + d CE ) . H ΦΨ Proof. It suffices to show that (cid:88) k ≥ φ ( ρh ) k ρψ = d CE : ∧ p ( L /K ) ∨ ⊗ V r → ∧ p +1 ( L /K ) ∨ ⊗ V r , where φ, ψ and h are defined in Lemma 4.10 and ρ = ˜ l ∨ + ˜ l ∨ + l ∨ .In fact, since ˜ l ∨ ◦ ψ : ∧ p ( L /K ) ∨ ⊗ V r → ∧ p +1 ( L ) ∨ ⊗ V r , by the definition of ˜ l , and l ∨ : ∧ p ( L /K ) ∨ ⊗ V r → ∧ p +1 ( L ) ∨ ⊗ V r , OHOMOLOGY OF HEMISTRICT LIE 2-ALGEBRAS 21 by the assumption that ι α l ∨ = l α = 0 for all α ∈ L − , it follows that hρψ = h ( − ˜ l ∨ + ˜ l ∨ + l ∨ ) ψ = h (˜ l ∨ + l ∨ ) ψ = 0 : ∧ p ( L /K ) ∨ ⊗ V r → C p + r ( ˜ L, V ) . Thus, (cid:88) k ≥ φ ( ρh ) k ρψ = φρψ = φ ( − ˜ l ∨ + ˜ l ∨ + l ∨ ) ψ = φ ( − ˜ l ∨ + l ∨ ) ψ = d CE : ∧ p ( L /K ) ∨ ⊗ V r → ∧ p +1 ( L /K ) ∨ ⊗ V r . To see the reason why the last equality holds, it suffices to prove the case p = 1: We compute for each ξ ∈ Hom( L /K, V r ), x, y ∈ L such that ¯ x = pr( x ) , ¯ y = pr( y ) ∈ L /K , φ (˜ l ∨ ( ψ ( ξ )))(¯ x, ¯ y ) = ξ (pr(˜ l ( j (¯ x ) , j (¯ y )))) = 12 ξ (pr([ j (¯ x ) , j (¯ y )] − [ j (¯ y ) , j (¯ x )]))= 12 ξ (pr([ x, y ] − [ y, x ])) = 12 ξ (pr([ x, y ] − [ y, x ]) + pr( dh ( x, y )))= 12 ξ (pr([ x, y ] − [ y, x ]) + pr([ x, y ] + [ y, x ])) = ξ ( { ¯ x, ¯ y } ) , and φ ( l ∨ ( ψ ( ξ )))(¯ x, ¯ y ) = l j (¯ x ) ξ (¯ y ) − l j (¯ y ) ξ (¯ x ) , which implies that φ ( l ∨ ( ψ ( ξ ))) − φ (˜ l ∨ ( ψ ( ξ ))) = d CE ( ξ ) as desired. (cid:3) As an immediate consequence, we have H • CE ( ˜ L, V ) ∼ = H • CE ( L Lie , V ), which completes the proof of Theo-rem 4.4.4.3. Application: Leibniz algebras. Let ( g , [ − , − ] g ) be a Leibniz algebra with Leibniz kernel K . ByExample 2.17, we have an injective hemistrict Lie 2-algebra L g := ( K [1] (cid:44) → g , [ − , − ] g , h ) . The associated Lie algebra is commonly denoted by g Lie . It is clear that L g satisfies the assumptions inTheorem 4.4. Note also that the alternator h is surjective in this case. In fact, any injective hemistrict Lie2-algebra with surjective alternator h is of this form (cf. [18]).Meanwhile, according to Roytenberg [18], Sheng and Liu [21], the skew-symmetrization of the Leibniz bracket[ − , − ] g gives rise to a semistrict Lie 2-algebra G := ( K [1] (cid:44) → g , (cid:101) l , (cid:101) l ) , where (cid:101) l is the skew-symmetrization of [ − , − ] g , i.e, (cid:101) l ( x, y ) = 12 ([ x, y ] g − [ y, x ] g ) , (cid:101) l ( x, α ) = − (cid:101) l ( α, x ) = 12 [ x, α ] g , for all x, y ∈ g , α ∈ K [1], and (cid:101) l : ∧ g → K [1] is defined by (cid:101) l ( x, y, z ) = − 112 ( h ([ x, y ] g − [ y, x ] g , z ) + h ([ y, z ] g − [ z, y ] g , x ) + h ([ z, x ] g − [ x, z ] g , y ))= 14 ([[ z, y ] g , x ] g + [[ x, z ] g , y ] g + [[ y, x ] g , z ] g ) . Applying Theorem 4.4, we have the following Theorem 4.12. Let ( g , [ − , − ] g ) be a Leibniz algebra with Leibniz kernel K and V a representation of L g such that l α = 0 for all α ∈ K . Then H • ( L g , V ) ∼ = H • ( G , V ) ∼ = H • CE ( g Lie , V ) . 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Chern Institute of Mathematics, Nankai University, Tianjin E-mail address : [email protected] Department of Mathematics, Peking University, Beijing E-mail address : [email protected] Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan E-mail address ::