Leibniz algebra



Sometimes in the place where we expect Lie algebras, some noncommutative phenomena occur and we need to drop out the requirement of antisymmetry of the brackets.

Jean-Louis Loday introduced Leibniz algebras, because of considerations in algebraic K-theory. Roughly speaking the Lie algebra homology is related to the appearance of cyclic homology (as it is manifest in the original work of Tsygan and then of Loday-Quillen).

Lie algebra homology involves the Chevalley-Eilenberg chain complex, which in turns involves the exterior powers of the Lie algebra. Loday found that there is a noncommutative generalization where roughly speaking one has the tensor and not the exterior powers of the Lie algebra in the complex; this new complex defines the Leibniz homology of Lie algebras. The Leibniz homology is related to the Hochschild homology the same way the Lie algebra homology is related to the cyclic homology.

  • C. Cuvier, Homologie de Leibniz et homologie de Hochschild, C.R. Acad. Sci. Paris, Ser. A-B313, 569-572 (1991)

In fact this new complex for Leibniz homology further generalizes to the case of Leibniz algebras, where it computes certain Tor groups for corepresentations of Leibniz algebras.


Given a commutative unital ring kk (usually a field), a Lebniz kk-algebra AA is a particular kind of nonassociative algebra over kk which is somewhat more general than a Lie algebra over kk.

A left Leibniz kk-algebra is kk-module LL equipped with a bracket, which is a kk-linear map [,]:AAA[,]:A\otimes A \to A satisfying the left Leibniz identity

[a,[b,c]]=[[a,b],c]+[b,[a,c]] [a, [b,c]] = [[a,b],c]+[b,[a,c]]

In other words, the left adad-map, a(ad la=[a,]:LL)a \mapsto (ad_l a = [a,-]:L\to L) is a derivation of LL as a nonassociative algebra. Similarly, there are right Leibniz algebras, for which the right adad-map ad r:a[,a]:LLad_r :a\mapsto [-,a]:L\to L is a derivation. In the presence of antisymmetry, the left Leibniz identity is equivalent to the Jacobi identity, though this is not true in general; thus a Lie algebra is precisely an antisymmetric (or alternating) Leibniz algebra.

Relation to Lie algebras in Loday-Pirashvili category

There is a remarkable observation of Loday and Pirashvili that in the Loday–Pirashvili tensor category of linear maps with (exotic) “infinitesimal tensor product”, the category of internal Lie algebras has the category of, say left, Leibniz kk-algebras as a full subcategory.


Some people dislike the term (left/right) Leibniz algebra (which is allegedly due to Loday), and prefer other names, including ‘Loday algebras’ and many longer descriptive names.

Corepresentation, representation, crossed module

Both a representation and a corepresentation of a right Leibniz kk-algebra 𝔤\mathfrak{g} involve a kk-module MM and two kk-linear maps “actions” M𝔤MM\otimes\mathfrak{g}\to M and 𝔤MM\mathfrak{g}\otimes M\to M with 3 axioms.

For representations:

[m,[x,y]]=[[m,x],y][[m,y],x][m, [x, y]] = [[m, x], y] - [[m, y], x]
[x,[a,y]]=[[x,m],y][[x,y],m][x, [a, y]] = [[x, m], y] - [[x, y], m]
[x,[y,m]]=[[x,y],m][[x,m],y][x, [y, m]] = [[x, y], m] - [[x, m], y]

for x,y𝔤x,y\in\mathfrak{g} and for mMm\in M.

For corepresentatons:

[[x,y],m]=[x,[y,m]][y,[x,m]][[x, y], m] = [x, [y, m]] - [y, [x, m]]
[y,[a,x]]=[[y,m],x][m,[x,y]][y, [a, x]] = [[y, m], x] - [m, [x, y]]
[[m,x],y]=[m,[x,y]][[y,m],x].[[m, x], y] = [m, [x, y]] - [[y, m], x].

If the two “actions” are symmetric, i.e. [x,m]+[m,x]=0[x,m] + [m,x] = 0 for all mMm\in M, x𝔤x\in\mathfrak{g} then all the 6 axioms of representation or corepresentation are equivalent. If MM is underlying a Leibniz algebra then an action of 𝔤\mathfrak{g} on MM is by definition symmetric, hence all the 6 equivalent conditions hold.

A map t:𝔤𝔟t : \mathfrak{g}\to\mathfrak{b} together with an action of 𝔟\mathfrak{b} on 𝔤\mathfrak{g} is a Leibniz crossed module if

t([b,g])=[b,t(g)],t([g,b])=[t(g),b],forallb𝔟,g𝔤 t([b,g])= [b,t(g)],\,\,\,t([g,b])=[t(g),b],\,\,\,\, for all\,\,\, b\in\mathfrak{b}, g' \in\mathfrak{g}
[g,t(g)]=[g,g]=[t(g),g],forallg,g𝔤 [g, t(g')] = [g, g'] = [t(g), g'],\,\,\,\, for all\,\,\, g, g' \in\mathfrak{g}

Abelian extensions

Abelian extension of right Leibniz algebras is a split short exact sequence of kk-modules

0M𝔥𝔤0 0\to M \to \mathfrak{h}\to \mathfrak{g}\to 0

where the mapping 𝔥𝔤\mathfrak{h}\to\mathfrak{g} is a morphism of Leibniz algebras, and MM is equipped with induced action of 𝔤\mathfrak{g}. The isomorphisms of extensions of 𝔤\mathfrak{g} by MM with fixed action are defined as usual. This way we obtain a set of equivalence classes Ext(𝔤,M)Ext(\mathfrak{g},M). To classify the extensions one looks for compatible Leibniz brackets on M𝔤M\oplus \mathfrak{g}. The general form of a bracket is

[(m 1,x 1),(m 2,x 2)]=([m 1,x 2]+[x 1,m 2]+f(x 1,x 2),[x 1,x 2]), [(m_1,x_1),(m_2,x_2)] = ([m_1, x_2] + [x_1, m_2] + f(x_1, x_2), [x_1, x_2]),

where f(x 1,x 2)f(x_1,x_2) satisfy the following 2-cocycle identity:

[x,f(y,z)]+[f(x,z),y][f(x,y),z]=f([x,y],z)f([x,z],y)f(x,[y,z]) [x, f(y, z)] + [f(x, z), y] - [f(x, y), z] = f([x, y], z) - f([x, z], y) - f(x, [y, z])

The extension is split in the category of Leibniz algebras if ff is a boundary i.e. there exists a kk-module map g:𝔤Mg:\mathfrak{g}\to M such that

f(x,y)=[x,g(y)]+[g(x),y]g([x,y]),x,y,𝔤 f(x, y) = [x, g(y)] + [g(x), y] - g([x, y]), \,\,\,x,y,\in\mathfrak{g}

As for the Lie algebras, the group of abelian extensions agrees with the 2-cohomology HL 2(𝔤,M)HL^2(\mathfrak{g},M).

A kk-linear derivation of a right Leibniz algebra 𝔤\mathfrak{g} with values in its representation MM is a kk-linear map satisfying the Leibniz property with respect to the bracket:

δ([x,y])=[δ(x),y]+[x,δ(y)] \delta([x,y]) = [\delta(x),y]+[x,\delta(y)]

Such derivations form a kk-module Der(𝔤,M)Der(\mathfrak{g},M).

Homology and cohomology

The homology and cohomology of Leibniz algebra 𝔤\mathfrak{g} with abelian kk-module of coefficients, which is a corepresentation AA in the case of homology and a representation MM in the case of cohomology:

HL *(𝔤,A)=Tor * U𝔤(U(𝔤 Lie),A), HL_*(\mathfrak{g},A) = Tor^{U\mathfrak{g}}_*(U(\mathfrak{g}_{Lie}),A) ,
HL *(𝔤,M)=Ext U𝔤 *(U(𝔤 Lie),A) HL^*(\mathfrak{g},M) = Ext_{U\mathfrak{g}}^*(U(\mathfrak{g}_{Lie}),A)

where U(𝔤 Lie)U(\mathfrak{g}_{Lie}) is the universal enveloping of the maximal Lie algebra quotient 𝔤 Lie\mathfrak{g}_{Lie} of 𝔤\mathfrak{g} and U𝔤U\mathfrak{g} is the universal enveloping of a Leibniz algebra 𝔤\mathfrak{g}.

Fopr n0n\geq 0, the nn-cocycles are elements in C n(𝔤,M)=Hom k(𝔤 n,M)C^n(\mathfrak{g}, M) = Hom_k(\mathfrak{g}^{\otimes n}, M), satisfying the corresponding abelian cocycle condition determined by the differential

d n:C n(𝔤,M)C n+1(𝔤,M) d^n : C^n(\mathfrak{g}, M)\to C^{n+1}(\mathfrak{g}, M)
(d nf)(x 1,...,x n+1)=[x 1,f(x 2,,x n+1)]+ n+1 i=2(1) i[f(x 1,,x^ i,,x n+1),x i] (d^n f) (x_1, . . . , x_{n+1}) = [x_1,f(x_2,\ldots,x_{n+1})] +\sum_{n+1}^{i=2} (-1)^i [f(x_1,\ldots, \hat{x}_i, \ldots, x_{n+1}), x_i]

Notice a difference from the Lie algebra cocycles where instead of a tensor power we have an external power. Then HL *(𝔤,M)=H *(C *(𝔤,M),d *)HL^*(\mathfrak{g},M) = H^*(C^*(\mathfrak{g}, M),d^*).

There are standard interpretations of cocycles in low dimensions. For example for n=0n=0, HL 0(𝔤,M)HL^0(\mathfrak{g}, M) is the submodule of invariants. For n=1n=1 there is a natural projection Der(𝔤,M)HL 1(𝔤,M)Der(\mathfrak{g},M)\to HL^1(\mathfrak{g},M) whose kernel is generated by inner derivations.

The Leibniz operad is quadratic Koszul algebra whose Koszul dual operad is called the operad of dual Leibniz algebras or of Zinbiel algebras, see there.


A generalization of Lie integration to conjectural Leibniz groups has been conjectured by J-L. Loday. A local version via local Lie racks has been proposed in

  • Simon Covez, The local integration of Leibniz algebras, arXiv:1011.4112; On the conjectural cohomology for groups, arXiv:1202.2269; L’intégration locale des algèbres de Leibniz, Thesis (2010), pdf

This is partly based on earlier insights of Kinyon and Weinstein:

  • Michael K. Kinyon, Leibniz algebras, Lie racks, and digroups, J. Lie Theory 17:1 (2007) 099–114, arxiv:math.GR/0403509

  • Simon Covez, On the conjectural Leibniz cohomology for groups, Journal of K-theory 10:03, Dec 2012, pp 519-563 doi

Revised on December 13, 2013 08:05:37 by Zoran Škoda (