Differential graded Lie algebras and Leibniz algebra cohomology
aa r X i v : . [ m a t h . K T ] O c t DIFFERENTIAL GRADED LIE ALGEBRAS AND LEIBNIZ ALGEBRACOHOMOLOGY
JACOB MOSTOVOY
Abstract.
In this note, we interpret Leibniz algebras as differential graded Lie algebras. Namely, weconsider two functors from the category of Leibniz algebras to that of differential graded Lie algebras andshow that they naturally give rise to the Leibniz cohomology and the Chevalley-Eilenberg cohomology. Asan application, we prove a conjecture stated by Pirashvili in arXiv:1904.00121 [math.KT]. Introduction
Perhaps, the simplest non-trivial example of a differential graded (abbreviated as “DG”) Lie algebra is the cone on a Lie algebra g . It consists of two copies of g placed in degrees 0 and 1 with the differential of degree − g is isomorphic, as a differentialgraded U ( g )-module, to the Chevalley-Eilenberg complex of g which is a free resolution of the base fieldconsidered as a trivial U ( g )-module. The cone is, clearly, a functor from Lie algebras to DG Lie algebras;one may ask whether other such functors give rise to useful homology and cohomology theories in a similarway, with the corresponding universal enveloping algebras replacing the Chevalley-Eilenberg complex.The correct setting for this question may be that of the Leibniz algebras since they are, essentially, trun-cations of DG Lie algebras. In this note we observe that the complex calculating the Leibniz cohomology ofa Leibniz algebra g is also obtained from the universal enveloping algebra of a certain DG Lie algebra asso-ciated functorially with g . Moreover, we exhibit a Chevalley-Eilenberg complex for Leibniz algebras, whichreduces to the usual Chevalley-Eilenberg complex when the bracket of the Leibniz algebra is antisymmetric.In the case of trivial coefficients, the cohomology calculated by this Chevalley-Eilenberg complex is nothingmore but the cohomology of the maximal Lie quotient g Lie of g ; however, the complex itself is different fromthe Chevalley-Eilenberg complex of g Lie .We then use this interpretation of Leibniz homology in order to solve the conjecture of T. Pirashvili aboutthe vanishing of the homology of a certain graded Lie algebra complex associated with a free Leibniz algebra.The close connection between Leibniz algebras and DG Lie algebras is implicit in the works of J.-L. Lodayand T. Pirashvili. In particular, in [5], they show that Leibniz algebras can be thought of as Lie algebraobjects in the category of truncated chain complexes. There is a non-linear analogue of this relationship;namely, the correspondence between augmented racks and cubical monoids, see [1, 6].2.
Definitions
Differential graded Lie algebras.
Recall that a differential graded Lie algebra is a Lie algebra inthe tensor category of chain complexes. Explicitly, it is a graded vector space L = M i ∈ Z L i over a field k of characteristic zero, equipped with a bilinear bracket J − , − K : L i ⊗ L j → L i + j satisfying thegraded antisymmetry J x, y K = ( − | x || y | +1 J y, x K , and the graded Jacobi identity:( − | x || z | J x, J y, z KK + ( − | y || x | J y, J z, x KK + ( − | z || y | J z, J x, y KK = 0 , together with a differential d : L i → L i − which satisfies the graded Leibniz rule: d J x, y K = J dx, y K + ( − | x | J x, dy K . ere x , y and z are arbitrary homogeneous elements in L .The component L is a Lie algebra. A DG Lie algebra L is non-negatively graded if L i = 0 for i < Leibniz algebras.
A left Leibniz algebra g over a field k of characteristic zero is a vector space witha bilinear bracket [ − , − ] satisfying the left Leibniz identity[[ x, y ] , z ] = [ x, [ y, z ]] − [ y, [ x, z ]]for all x, y, z ∈ g . The definition of a right Leibniz algebra is similar, with the left Leibniz identity replacedwith the right Leibniz identity [ x, [ y, z ]] = [[ x, y ] , z ] − [[ x, z ] , y ] . If g is a left Leibniz algebra, the right Leibniz algebra g opp coincides with g as a vector space and has thebracket [ x, y ] g opp = [ y, x ] g . In the same fashion one defines the opposite of a right Leibniz algebra. We will mostly speak about leftLeibniz algebras and omit the term “left” when it cannot lead to confusion.The kernel of g is the ideal g ann linearly spanned by all elements of the form [ x, x ] with x ∈ g . Takingthe quotient by the kernel amounts to enforcing antisymmetry in g ; the Lie algebra g Lie := g / g ann is the maximal Lie quotient of g . The Lie algebra g Lie acts on g on the left: if d : g → g Lie denotes thequotient map, then, for each x ∈ g we have dx · y = [ x, y ] . The Leibniz functor and its adjoints
Enveloping DG Lie algebras of a Leibniz algebra.
For any DG Lie algebra L its degree one part L together with the bracket [ x, y ] := J dx, y K is a Leibniz algebra which we denote by Leib( L ). In what follows, we shall restrict our attention to non-negatively graded DG Lie algebras L such that L = (Leib( L )) Lie . These DG Lie algebras form a categorywhich we denote by
DGLie ; the functor
DGLie
Leib −−−→
Leib to the category of Leibniz algebras will be called the
Leibniz functor . Whenever g = Leib( L ), we shall saythat L is an enveloping DG Lie algebra of g . Lemma 1.
A DG Lie algebra L is in DGLie if and only if L → L is surjective and the kernel of thismap coincides with d J L , L K .Proof. Indeed, for x, y ∈ L we have d J x, y K = J dx, y K − J x, dy K = [ x, y ] Leib( L ) + [ y, x ] Leib( L ) , and, therefore, d J L , L K coincides with the kernel of the Leibniz algebra Leib( L ). (cid:3) The Leibniz functor has both left and right adjoint functors. The left adjoint functor was defined in [6].Consider g ⊕ g Lie as a chain complex of length two with g Lie in degree 0, g in degree 1 and the differential d : g → g Lie being the quotient map. The free graded Lie algebra on g ⊕ g Lie is a DG Lie algebra with thedifferential induced by d ; let E ( g ) be the quotient of this free DG Lie algebra by the relations(3.1) J x, y K = [ x, y ] g Lie when x, y ∈ g Lie , and(3.2) J x, y K = x · y when x ∈ g Lie and y ∈ g . We call E ( g ) the universal enveloping DG Lie algebra of g . The following isimmediate: called the derived bracket; see [2]. roposition 2. The universal enveloping DG Lie algebra functor is left adjoint to the Leibniz functor.
The right adjoint to the Leibniz functor is the minimal enveloping DG Lie algebra . It assigns to a Leibnizalgebra g the non-negatively graded 3-term DG Lie algebra M ( g ) defined as . . . → → → g ann i −→ g d −→ g Lie . Here, i is the inclusion, and the brackets in M ( g ) are defined as J a, b K = [ a, b ] g Lie for a, b ∈ g Lie , J a, x K = a · x when x ∈ g and a ∈ g Lie , i ( J a, x K ) = a · i ( x )for x ∈ g ann and a ∈ g Lie , and J x, y K = [ x, y ] g + [ y, x ] g when x, y ∈ g . It is a straightforward check that M ( g ) is, indeed a DG Lie algebra which is a functor of g . Proposition 3.
The functor g
7→ M ( g ) is right adjoint to the Leibniz functor.Proof. Consider a DG Lie algebra L = . . . → L d −→ L → L = L /d J L , L K . Define the homomorphism m : L → M (Leib( L )) by setting it to be the identity on L and L and, in degree 2, to coincide with d : L → Leib( L ) ann ⊆ L . Then, given a Leibniz algebra homomorphism φ : Leib( L ) → g the composition m ◦ M ( φ ) is the adjoint map. (cid:3) Remark 4.
In fact, the category of Leibniz algebras is equivalent to the subcategory of the acyclic DG Liealgebras in
DGLie which are zero in degrees three and higher, the equivalence being given by the functor M ( g ) and the Leibniz functor. Remark 5.
It is not hard to see that while the Leibniz functor could be defined on the category
DGLie of all DG Lie algebras, it would fail to have the right adjoint there. In particular, given a homomorphism
Leib( L ) → g there is no canonical way to define the action of the Lie algebra L on g which is part of thestructure of an adjoint functor. This can be resolved by choosing L to coincide with (Leib( L )) Lie , that is,considering the category
DGLie . Remark 6.
The terms “universal enveloping” and “minimal enveloping” for the DG Lie algebras E ( g ) and M ( g ) reflect the fact that, in the category of the enveloping DG Lie algebras of a given Leibniz algebra g ,the DG Lie algebra E ( g ) is the initial, and M ( g ) is the terminal object. Representations of Leibniz algebras as DG modules.
A chain complex M is a (DG) module overa DG Lie algebra L if it equipped with a degree zero bracket J − , − K : L ⊗ M → M, which satisifes JJ x, y K , m K = J x, J y, m KK − ( − | y || x | J y, J x, m KK and d J x, m K = J dx, m K + ( − | x | J x, dm K . Here x and y are arbitrary homogeneous elements in L and m is a homogeneous element of M .This is the definition of a left module; any left module over a DG Lie algebra is also a right module withthe bracket J m, x K = ( − | x || m | +1 J x, m K . A representation of a left Leibniz algebra g is a vector space m equipped with bilinear brackets g ⊗ m → m and m ⊗ g → m satisfying [[ m, x ] , y ] = [ m, [ x, y ]] − [ x, [ m, y ]] , [[ x, m ] , y ] = [ x, [ m, y ]] − [ m, [ x, y ]] , [[ x, y ] , m ] = [ x, [ y, m ]] − [ y, [ x, m ]] , here x, y ∈ g and m ∈ m .A representation m of a right Leibniz algebra g is defined in the similar way; instead of the left, it satisfiesthe right Leibniz identity where two of the arguments lie in g and one lies in m . Loday and Pirashvili use theterm “co-representation” for a representation of the opposite Leibniz algebra. Given a representation m of g ,the representation m opp of g opp is defined as the same vector space as m with the brackets [ x, m ] m opp = [ m, x ] m and [ m, x ] m opp = [ x, m ] m .For any L -module M , the space M i becomes a representation of Leib( L ) if we define(3.3) [ x, m ] = J dx, m K and(3.4) [ m, x ] = − J x, dm K for x ∈ L and m ∈ M i .Given a representation m of a Leibniz algebra g , denote by m anti ⊆ m the subspace spanned by all theexpressions of the form [ x, m ] + [ m, x ] and let m symm be the quotient m / m anti . Consider the quotient map d : m → m symm as the differential in the DG vector space whose only non-trivial components are m in degreezero and m symm in degree − E ( m ) as the quotient of the free E ( g )-module generated by the DG vector space m → m symm bythe relations (3.3) and (3.4). As a graded (not DG) vector space, E ( m ) := E ( m ) ≥ coincides with the freemodule over the free graded Lie algebra generated by g in degree 1. In particular, for i ≥ E i ( m ) = m ⊗ g ⊗ i . The differential E ( m ) → E − ( m ) coincides with m → m symm .Consider the DG vector space M ( m ) := . . . → → m anti i −→ m d −→ m symm → → . . . with m in degree zero, i the inclusion and d the quotient map. Proposition 7.
The DG vector space M ( m ) has a natural structure of an M ( g ) -module.Proof. Let us construct an explicit action of an arbitrary enveloping DG Lie algebra L of g on M ( m ).The last of the three conditions satisfied by a representation implies that[[ x, y ] + [ y, x ] , m ] = 0and therefore, the bracket g ⊗ m → m descends to a Lie algebra action g Lie ⊗ m → m . This action preserves m anti since [ x, [ y, m ]] + [ x, [ m, y ]] = [[ x, y ] , m ] + [ m, [ x, y ]] + [ y, [ x, m ]] + [[ x, m ] , y ]and, therefore, descends to an action of g Lie on m symm . These three actions together give the action of L on M ( m ).Define the maps L ⊗ M i ( m ) → M i +1 ( m ), i = − , x ⊗ dm
7→ − [ m, x ] ,x ⊗ m [ x, m ] + [ m, x ]and the map L ⊗ M − ( m ) → M ( m ) by J x, y K ⊗ dm
7→ − [ m, [ x, y ] + [ y, x ]] . Verifying that M ( m ) is indeed an L -module is straightforward. (cid:3) Remark 8.
The choice of grading on a DG module over a DG Lie algebra L is arbitrary, since a shift ingrading produces another DG module over L . Note that a certain ambiguity in our notation: if g is consideredas a representation of itself, the DG module M ( g ) is obtained from the DG Lie algebra M ( g ) by shifting thegrading down by one. . Leibniz algebra homology and cohomology via enveloping DG Lie algebras
Homology and cohomology for right Leibniz algebras were defined by Loday; see [3, 4] . Here, we areconcerned with left Leibniz algebras; by the cohomology HL ∗ ( g , m ) of a left Leibniz algebra g with coefficientsin a representation m we understand the cohomology of the right Leibniz algebra g opp , with coefficients in m opp ; similarly, the homology HL ∗ ( g , m ) of g is the homology of g opp with coefficients in m .4.1. The Chevalley-Eilenberg complex of a Leibniz algebra.
For any Leibniz algebra g , its minimalenveloping DG Lie algebra M ( g ) is acyclic. As a consequence, the universal enveloping algebra U ( M ( g )) hasno homology in positive dimensions (see Proposition 2.1 of [8, Appendix B]). It can be seen that the gradedcomponents of U ( M ( g )) are free as left U ( g Lie )-modules. Therefore, U ( M ( g )) is a free resolution of thebase field k considered as a trivial U ( g Lie )-module; we call it the
Chevalley-Eilenberg complex of the Leibnizalgebra g . When g is a Lie algebra, M ( g ) is the cone on g and U ( M ( g )) is the usual Chevalley-Eilenbergcomplex. Proposition 9.
Let m be a module over the Lie algebra g Lie . The chain complex m ⊗ U ( M ( g )) calculatesthe homology, and the cochain complex Hom( U ( M ( g )) , m ) the cohomology of the maximal Lie quotient g Lie with coefficients in m . Here, the tensor product and Hom are taken in the category of g Lie -modules.Proof.
It only remains to see that the graded components of U ( M ( g )) are free as left U ( g Lie )-modules.Indeed, U ( M ( g )), as a graded left U ( g Lie )-module, is isomorphic to U ( g Lie ) ⊗ U ( M ( g ) > ), where M ( g ) > isthe graded Lie algebra obtained from M ( g ) by setting the zero degree component to be trivial. Therefore,each graded component of U ( M ( g )) is free. (cid:3) Leibniz homology and cohomology with coefficients in a Lie algebra representation.
Onecan generalise the definition of the Chevalley-Eilenberg complex by replacing the minimal enveloping DGLie algebra M ( g ) with an arbitrary enveloping DG Lie algebra functor. The resulting complex will notnecessarily be acyclic, of course; however, it may produce interesting functors of g instead of the usualhomology and cohomology. In particular, one can consider the universal enveloping DG Lie algebra E ( g ). Proposition 10.
Let m be a module over the Lie algebra g Lie . The chain complex m ⊗ U ( E ( g )) calculates theLeibniz homology, and the cochain complex Hom( U ( E ( g )) , m ) the Leibniz cohomology of g with coefficientsin m . Here, the tensor product and Hom are taken in the category of g Lie -modules.Proof.
The universal enveloping algebra U ( E ( g )) is isomorphic, as a left U ( g Lie )-module, to U ( g Lie ) ⊗ k T ( g ) . Indeed, it is the tensor algebra on g ⊕ g Lie modulo the relations ax − xa = a · x for a ∈ g Lie and x ∈ g . Here,we write the product in T ( g ) simply as juxtaposition. Therefore, as left U ( g Lie )-modules, m ⊗ g Lie U ( E ( g )) = m ⊗ k T ( g ) . The differential in m ⊗ k T ( g ) is induced by d : g → g Lie ; in particular, for u ∈ m and x , . . . , x n ∈ g wehave d ( u ⊗ x x . . . x n ) = u ⊗ d ( x ) x . . . x n − u ⊗ x d ( x ) . . . x n + . . . + ( − n +1 u ⊗ x x . . . d ( x n )= X ≤ i Let L be a non-negatively graded DG Lie algebra and M a DG module over L . Write M for the chaincomplex obtained from M by replacing each M i with i < M may fail to be a DG moduleover L ; however, it is a graded module over L considered as a graded (not DG) Lie algebra.In order to extend the observations of the previous subsections to the more general case of coefficients ina Leibniz algebra representation, we replace the module m by E ( m ) or M ( m ) and the category of g -modulesby the category of graded, although not differential graded, E ( g )-modules. Proposition 11. Let m be a representation of a Leibniz algebra g . The chain complex E ( m ) ⊗ U ( E ( g )) calculates the Leibniz homology, and the cochain complex Hom( U ( E ( g )) , M ( m )) the Leibniz cohomology of g with coefficients in m . Here, the tensor product and Hom are taken in the category of graded E ( g ) -modules.Proof. As we have already noted before, E ( m ) is the free module over the free graded Lie algebra generatedby g in degree one. Therefore, the chain complex E ( m ) ⊗ U ( E ( g )) is isomorphic to m ⊗ T ( g ); one verifiesdirectly that it is the standard complex for the Leibniz homology of g opp with coefficients in m .The n th term of the cochain complex Hom( U ( E ( g )) , M ( m )) consists of E ( g )-module morphisms of degree − n , with the differential df = f ◦ d − ( − deg f d ◦ f. It can be identified with the space of linear maps Hom k ( g ⊗ n , m ), since M ( m ) is generated in degree zeroby m . Indeed, let f i be the restriction of f ∈ Hom( U ( E ( g )) , M ( m )) to g ⊗ i . Then, given f n : g ⊗ n → m the E ( g )-module morphism f is reconstructed by applying the left action of E ( g ). In particular, taking thetensor product with (the identity morphism of) g ∈ E ( g ) we obtain the map f n +1 : g ⊗ ( n +1) → M ( m ) x . . . x n +1 ( − n J x , f n ( x , . . . , x n +1 ) K and, therefore,( − n d ◦ f = ( − n ( J dx , f n ( x , . . . , x n +1 ) K − J x , df n ( x , . . . , x n +1 ) K )= [ x , f n ( x , . . . , x n +1 )] + [ f n ( x , . . . , x n +1 ) , x ] , which is the correction term to (4.1) in the case of coefficients in a Leibniz algebra representation, see[3, 4]. (cid:3) Chevalley-Eilenberg homology and cohomology with coefficients in a Leibniz algebra rep-resentation. Consider the chain complex E ( m ) ⊗ E ( g ) U ( M ( g ))and the cochain complex Hom E ( g ) ( U ( M ( g )) , M ( m )) . When m is a representation of g Lie , these complexes coincide with m ⊗ g Lie U ( M ( g )) and Hom g Lie ( U ( M ( g )) , m ),respectively.Therefore, we may denote the respective homology and cohomology groups by H ∗ ( g , m ) and H ∗ ( g , m ). Proposition 12. The groups H ( g , m ) and H ( g , m ) coincide with the respective Leibniz (co)homologygroups HL ( g , m ) and HL ( g , m ) . The natural map HL ( g , m ) → H ( g , m ) is surjective while H ( g , m ) → HL ( g , m ) is injective. roof. Indeed, the quotient E ( g ) → M ( g ) is the identity in degrees 0 and 1. In degree 2 we have E ( g ) = S ( g )with d ( x, y ) = [ x, y ] + [ y, x ], while M ( g ) = g ann and d : M ( g ) → M ( g ) = g is injective. Since S ( g ) → g ann is surjective, the quotient map induces an isomorphism in the homology in degrees 0 and 1and the same is true for the induced maps E ( m ) ⊗ U ( E ( g )) → E ( m ) ⊗ U ( M ( g ))and Hom( U ( E ( g )) , M ( m )) → Hom( U ( M ( g )) , M ( m )) . The surjectivity of the homology and the injectivity of the cohomology in degree 2 follows from the surjectivityof the quotient map in degree 2 and the injectivity of g ann → g . (cid:3) The free Lie algebra complex Consider the graded tensor algebra T ( V ) generated by a vector space V in degree 1. Inside T ( V ), thevector space V generates the free graded Lie algebra F ( V ) by means of the operation of the (graded)commutator J x, y K = xy − ( − | x || y | yx. For a Leibniz algebra g , the graded Lie algebra F ( g ) is a subcomplex of the standard complex T ( g ) thatcalculates the Leibniz homology of g . Pirashvili in [7] conjectures the following: Proposition 13. If the Leibniz algebra g is free, the homology of F ( g ) vanishes in dimensions greater thanone.Proof. The components of positive degree of the universal enveloping DG Lie algebra E ( g ) form the freegraded Lie algebra E > ( g ) on g . We claim that the differential on E > ( g ) induced from E ( g ) coincides withthe one coming from the Leibniz complex T ( g ). Indeed, we have natural maps of DG vector spaces E ( g ) → U ( E ( g )) → k ⊗ U ( g Lie ) U ( E ( g )) = T ( g )whose composition identifies E > ( g ) with F ( g ).Now, assume that g is a free Leibniz algebra on the set X and consider the graded vector space W = . . . → → → W X id −→ W X , where W X is the vector space spanned by X , placed in degrees 0 and 1. It follows from the definitions that E ( g ) can be identified with the free DG Lie algebra ( F ( W ) , d ) generated by W .By Proposition 2.1 of [8, Appendix B], since W is acyclic, the tensor algebra T ( W ) has no homologyin positive dimensions and, therefore, the free DG Lie algebra E ( g ) on W is also acyclic; this implies thestatement of the Proposition. (cid:3) References [1] F. Clauwens, The algebra of rack and quandle cohomology, Journal of Knot Theory and Its Ramifications bf 20 (2011)1487–1535.[2] Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras Annales de l’institut Fourier, (1996) 1243–1274.[3] J.-L. Loday, Une version non commutative des alg`ebres de Lie: Les alg`ebres de Leibniz , Enseign. Math., II. S´er. , No.3-4 (1993), 269–293.[4] J.-L. Loday, T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology , Math. Ann. (1993)139–158.[5] J.-L. Loday, T. Pirashvili, The tensor category of linear maps and Leibniz algebras,