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.
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The Leibniz´s identity is best 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](../../../../math/4/7/1/471055de4695b84817bb82931cd8cf8b.png)
) 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.
