In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, 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 $a,b],c]\; =\; [a,[b,c+\; derivation]].\; If\; in\; addition\; the\; bracket\; is\; alternating\; ([$a, a] = 0) then the Leibniz algebra is a [[Lie algebra]]. Indeed, in this case [a, b] = −[b, a] and the Leibniz's 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. The tensor module, T(V) , of any vector space V can be turned into a Loday algebra such that $[a\_1\backslash otimes\; \backslash cdots\; \backslash otimes\; a\_n,x]=a\_1\backslash otimes\; \backslash cdots\; a\_n\backslash otimes\; x\backslash quad\; \backslash text\{for\; \}a\_1,\backslash ldots,\; a\_n,x\backslash in\; V.$ This is the free Loday algebra over V. Leibniz algebras were discovered by Jean-Louis Loday by notice 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 Leibniz homology of L is the tensor algebra over the Hochschild homology

Hochschild homology

In mathematics, Hochschild homology is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors...

of A.