Leibniz algebra

 ( Lie Algebras ) 

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

$[a,b,c]\; =\; a,[b,c]+\; [a,c,b].\; \backslash ,$

In other words, right multiplication by any element c is a derivation. If in addition the bracket is alternating ( a,  a = 0) then the Leibniz algebra is a Lie algebra. Indeed, in this case a,  b = − b,  a and the Leibniz identity is equivalent to Jacobi's identity ( a, [ b,  c] +  c, [ a,  b] +  b, [ c,  a] = 0). 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 the Levi–Malcev theorem also holds.

The tensor module, T( V) , of any vector space V can be turned into a Loday algebra such that

$a\_1\backslash otimes=a\_1\backslash otimes\; \backslash cdots\; a\_n\backslash otimes\; x\backslash quad\; \backslash text\{for\; \}a\_1,\backslash ldots,\; a\_n,x\backslash in\; V.$

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( L) of this chain complex is known as Leibniz homology. 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.

A Zinbiel algebra is the Koszul duality concept to a Leibniz algebra. It has as defining identity:

$(\; a\; \backslash circ\; b\; )\; \backslash circ\; c\; =\; a\; \backslash circ\; (b\; \backslash circ\; c)\; +\; a\; \backslash circ\; (c\; \backslash circ\; b)\; .$

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Notes

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