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Leibniz algebra

Motivation

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).

• Jean-Louis Loday, Daniel Quillen, Cyclic homology and the Lie algebra homology of matrices, Comm. Math. Helv. 59, n. 1, 565-591 (1984), doi

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.

Definition

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

A left Leibniz $k$-algebra is $k$-module $L$ equipped with a bracket, which is a $k$-linear map $[,]:A\otimes A\to A$ 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 $\mathrm{ad}$-map, $a\mapsto ({\mathrm{ad}}_{l}a=[a,-]:L\to L)$ is a derivation of $L$ as a nonassociative algebra. Similarly, there are right Leibniz algebras, for which the right $\mathrm{ad}$-map ${\mathrm{ad}}_r:a\mapsto [-,a]:L\to L$ 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 $k$-algebras as a full subcategory.

Terminology

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 $k$-algebra $𝔤$ involve a $k$-module $M$ and two $k$-linear maps “actions” $M\otimes 𝔤\to M$ and $𝔤\otimes M\to M$ 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 $x,y\in 𝔤$ and for $m\in M$.

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. $[x,m]+[m,x]=0$ for all $m\in M$, $x\in 𝔤$ then all the 6 axioms of representation or corepresentation are equivalent. If $M$ is underlying a Leibniz algebra then an action of $𝔤$ on $M$ is by definition symmetric, hence all the 6 equivalent conditions hold.

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

$$t([b,g])=[b,t(g)],t([g,b])=[t(g),b],\mathrm{for}\mathrm{all}b\in 𝔟,g\prime \in 𝔤$$t([b,g])= [b,t(g)],\,\,\,t([g,b])=[t(g),b],\,\,\,\, for all\,\,\, b\in\mathfrak{b}, g' \in\mathfrak{g}

$$[g,t(g\prime )]=[g,g\prime ]=[t(g),g\prime ],\mathrm{for}\mathrm{all}g,g\prime \in 𝔤$$[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 $k$-modules

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

where the mapping $𝔥\to 𝔤$ is a morphism of Leibniz algebras, and $M$ is equipped with induced action of $𝔤$. The isomorphisms of extensions of $𝔤$ by $M$ with fixed action are defined as usual. This way we obtain a set of equivalence classes $\mathrm{Ext}(𝔤,M)$. To classify the extensions one looks for compatible Leibniz brackets on $M\oplus 𝔤$. 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 $f(x_1,x_2)$ 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 $f$ is a boundary i.e. there exists a $k$-module map $g:𝔤\to M$ such that

$$f(x,y)=[x,g(y)]+[g(x),y]-g([x,y]),x,y,\in 𝔤$$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 ${\mathrm{HL}}^2(𝔤,M)$.

A $k$-linear derivation of a right Leibniz algebra $𝔤$ with values in its representation $M$ is a $k$-linear map satisfying the Leibniz property with respect to the bracket:

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

Such derivations form a $k$-module $\mathrm{Der}(𝔤,M)$.

Homology and cohomology

The homology and cohomology of Leibniz algebra $𝔤$ with abelian $k$-module of coefficients, which is a corepresentation $A$ in the case of homology and a representation $M$ in the case of cohomology:

$${\mathrm{HL}}_{*}(𝔤,A)={\mathrm{Tor}}_{*}^{U𝔤}(U({𝔤}_{\mathrm{Lie}}),A),$$HL_*(\mathfrak{g},A) = Tor^{U\mathfrak{g}}_*(U(\mathfrak{g}_{Lie}),A) ,

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

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

Fopr $n\ge 0$, the $n$-cocycles are elements in $C^n(𝔤,M)={\mathrm{Hom}}_k({𝔤}^{\otimes n},M)$, satisfying the corresponding abelian cocycle condition determined by the differential

$$d^n:C^n(𝔤,M)\to C^{n+1}(𝔤,M)$$d^n : C^n(\mathfrak{g}, M)\to C^{n+1}(\mathfrak{g}, M)

$$(d^{n}f)(x_1,...,x_{n+1})=[x_1,f(x_2,\dots ,x_{n+1})]+\sum _{n+1}^{i=2}(-1{)}^i[f(x_1,\dots ,{\hat{x}}_i,\dots ,x_{n+1}),x_i]$$(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 ${\mathrm{HL}}^{*}(𝔤,M)=H^{*}(C^{*}(𝔤,M),d^{*})$.

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

The related kinds of algebras

The Leibniz operad is quadratic. The Koszul dual of the Leibniz operad is called the operad of dual Leibniz algebras. A dual Leibniz algebra (in $\mathrm{Vec}$) is a nonassociative algebra with a product satisfying

$$(xy)z=x(zy+yz),$$(x y) z = x (z y + y z),

It is of course natural to consider also dg-algebras over a dg-version of the operad of dual Leibniz algebras.

Literature

• Jean-Louis Loday, Teimuraz Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann. 296, 139-158 (1993), pdf

• Jean-Louis Loday, Teimuraz Pirashvili, The tensor category of linear maps, Georg. Math. J. vol. 5, n.3 (1998) 263–276.

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