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  In mathematics, a (right) Leibniz algebra, named after Gottfried Wilhelm Leibniz, sometimes called a Loday algebra, after JeanLouis Loday, is a module L over a commutative ring R with a bilinear product [ _ , _ ] satisfying the Leibniz identity The tensor module, T(V) , of any vector space V can be turned into a Loday algebra such that This is the free Loday algebra over V. A Zinbiel algebra is the Koszul dual concept to a Leibniz algebra. It has defining identity:

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  In mathematics, a (right) Leibniz algebra, named after Gottfried Wilhelm Leibniz, sometimes called a Loday algebra, after JeanLouis 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 ([a, a] = 0) then the Leibniz algebra is a Lie algebra. Indeed, in this case [a, b] = −[b, a] and the Leibniz's identity is equivalent to Jacobi's identity ([a, [b, c]] + [c, [a, b]] + [b, [c, a]] = 0). Conversely any Lie algebra is obviously a Leibniz algebra. In this sense, Leibniz algebras can be seen as a noncommutative 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 LeviMalcev theorem also holds. The tensor module, T(V) , of any vector space V can be turned into a Loday algebra such that This is the free Loday algebra over V. Leibniz algebras were discovered in 1965 by A. Bloh, who called them Dalgebras. They attracted interest after JeanLouis Loday noticed that the classical in the exterior module of a Lie algebra can be lifted to the tensor module which yields a new chain complex. In fact this complex is welldefined for any Leibniz algebra. The homology HL(L) of this chain complex is known as . If L is the Lie algebra of (infinite) matrices over an associative Ralgebra A then Leibniz homologyof L is the tensor algebra over the Hochschild homology of A. A Zinbiel algebra is the Koszul dual concept to a Leibniz algebra. It has defining identity:

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