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.
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.
A left Leibniz -algebra is -module equipped with a bracket, which is a -linear map satisfying the left Leibniz identity
In other words, the left -map, is a derivation of as a nonassociative algebra. Similarly, there are right Leibniz algebras, for which the right -map 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.
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 -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.
Both a representation and a corepresentation of a right Leibniz -algebra involve a -module and two -linear maps “actions” and with 3 axioms.
for and for .
If the two “actions” are symmetric, i.e. for all , then all the 6 axioms of representation or corepresentation are equivalent. If is underlying a Leibniz algebra then an action of on is by definition symmetric, hence all the 6 equivalent conditions hold.
A map together with an action of on is a Leibniz crossed module if
Abelian extension of right Leibniz algebras is a split short exact sequence of -modules
where the mapping is a morphism of Leibniz algebras, and is equipped with induced action of . The isomorphisms of extensions of by with fixed action are defined as usual. This way we obtain a set of equivalence classes . To classify the extensions one looks for compatible Leibniz brackets on . The general form of a bracket is
where satisfy the following 2-cocycle identity:
The extension is split in the category of Leibniz algebras if is a boundary i.e. there exists a -module map such that
As for the Lie algebras, the group of abelian extensions agrees with the 2-cohomology .
A -linear derivation of a right Leibniz algebra with values in its representation is a -linear map satisfying the Leibniz property with respect to the bracket:
Such derivations form a -module .
The homology and cohomology of Leibniz algebra with abelian -module of coefficients, which is a corepresentation in the case of homology and a representation in the case of cohomology:
where is the universal enveloping of the maximal Lie algebra quotient of and is the universal enveloping of a Leibniz algebra .
Fopr , the -cocycles are elements in , satisfying the corresponding abelian cocycle condition determined by the differential
Notice a difference from the Lie algebra cocycles where instead of a tensor power we have an external power. Then .
There are standard interpretations of cocycles in low dimensions. For example for , is the submodule of invariants. For there is a natural projection 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.
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