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In mathematics, a 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
In other words, right multiplication by any element c is a derivation. If in addition the bracket is alternating = 0] then the Leibniz algebra is a Lie algebra. Indeed, in this case = − and the Leibniz's identity is equivalent to Jacobi's identity ] + ] + ] = 0]. Conversely any Lie algebra is obviously a Leibniz algebra.
In this sense, Leibniz algebras can be seen as a non-commutative generalization of Lie algebras. The investigation of which theorems and properties of Lie algebras are still valid for Leibniz algebras is a recurrent theme in the literature. For instance, it has been shown that Engel's theorem still holds for Leibniz algebras and that a weaker version of Levi-Malcev theorem also holds.
The tensor module, T , of any vector space V can be turned into a Loday algebra such that