Leibniz algebra

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In mathematics, a (left) Leibniz algebra (sometimes called a Loday algebra) is a module A over a commutative ring 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.

[edit] References

• Springer encyclopedia article
• J.-L. Loday, Cyclic homology, Springer (1992) (Second ed.: 1998)
• J.-L. Loday, Une version non commutative des algèbres de Lie: les algèbres de Leibniz Enseign. Math. , 39 (1993) pp. 269–293
• J.-L. Loday, Overview on Leibniz algebras, dialgebras and their homology Fields Inst. Comm. , 17 (1997) pp. 91–102