Leibniz algebra

In mathematics, a (left) Leibniz algebra (sometimes called a Loday algebra) is a module "A" 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 anticonmutative (i.e. $[a,b]\; =\; -\; [b,a]\; ;\; equiv\; ;\; [a,a]\; =0$) 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.

References

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