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algebra > differential algebra > Lie algebra > Leibniz algebra

... > abstract algebra > algebraic structure > algebra over a field > Lie algebra > Leibniz algebra

Preferred term

Leibniz algebra

Definition(s)

In mathematics, a (right) 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
${\\displaystyle [[a,b],c]=[a,[b,c]]+[[a,c],b].\\,}$
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.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Leibniz_algebra)

Broader concept(s)

Synonym(s)

Loday algebra

In other languages

algèbre de Leibniz

French
algèbre de Loday

URI

http://data.loterre.fr/ark:/67375/PSR-PGB1XCV4-R

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