In mathematics, a (right) 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.
[1] For instance, it has been shown that Engel's theorem still holds for Leibniz algebras[2][3] and that a weaker version of the Levi–Malcev theorem also holds.
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.
If L is the Lie algebra of (infinite) matrices over an associative R-algebra A then the Leibniz homology of L is the tensor algebra over the Hochschild homology of A.