Leibniz algebra


In mathematics, a Leibniz algebra, named after Gottfried Wilhelm Leibniz, sometimes called a Loday algebra, after Jean-Louis Loday, is a module L over a commutative ring R with a bilinear product satisfying the Leibniz identity
In other words, right multiplication by any element c is a derivation. If in addition the bracket is alternating then the Leibniz algebra is a Lie algebra. Indeed, in this case = − and the Leibniz's identity is equivalent to Jacobi's identity. Conversely any Lie algebra is obviously a Leibniz algebra.
In this sense, Leibniz algebras can be seen as a non-commutative generalization of Lie algebras. The investigation of which theorems and properties of Lie algebras are still valid for
Leibniz algebras is a recurrent theme in the literature. For instance, it has been shown that Engel's theorem still holds for Leibniz algebras and that a weaker version of Levi-Malcev theorem also holds.
The tensor module, T, of any vector space V can be turned into a Loday algebra such that
This is the free Loday algebra over V.
Leibniz algebras were discovered in 1965 by A. Bloh, who called them D-algebras. They attracted interest after Jean-Louis Loday noticed that the classical Chevalley–Eilenberg boundary map in the exterior module of a Lie algebra can be lifted to the tensor module which yields a new chain complex. In fact this complex is well-defined for any Leibniz algebra. The homology HL of this chain complex is known as Leibniz homology. If L is the Lie algebra of matrices over an associative R-algebra A then Leibniz homology
of L is the tensor algebra over the Hochschild homology of A.
A Zinbiel algebra is the Koszul dual concept to a Leibniz algebra. It has defining identity: