Leibniz algebra

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In mathematics, a (left) Leibniz algebra (sometimes called a Loday algebra) is a module L over a commutative ring or field R with a bilinear product [,] such that [a,[b,c]] = [[a,b],c] + [b,[a,c]]. In other words, left multiplication by any element a is a derivation.

If in addition the bracket is alternating ([a,a] = 0) then the Leibniz algebra is a Lie algebra. Conversely any Lie algebra is obviously a Leibniz algebra.

The Leibniz´s identity is also known by this formula: [a,[b,c]] = [[a,b],c] − [[a,c],b].

If in addition the bracket is anticommutative (i.e. ) then the Leibniz's identity is equivalent to Jacobi's identity ([a,[b,c]] + [c,[a,b]] + [b,[c,a]] = 0) and that's why in this case the Leibniz algebra is a Lie algebra.

Leibniz algebras were discovered by Jean-Louis Loday by notice that the classical Chevalley-Eilenberg boundary map in the exterior algebra of a Lie algebra has a canonical lifting to the tensor algebra 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 of A.

[edit] References

• Kosmann-Schwarzbach, Yvette (1996). "From Poisson algebras to Gerstenhaber algebras". Annales de l'institut Fourier 46, no. 5: 1243-1274. 
• Loday, Jean-Louis (1993). "Une version non commutative des algèbres de Lie: les algèbres de Leibniz". Enseign. Math. (2) 39, no. 3-4: 269-293.