Leibniz Algebra

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

: a,bc] = ,[b,c+__a,c.html"_;"title=",c.html"_;"title=",[b,c">,[b,c+__a,c">,c.html"_;"title=",,[b,c+__a,cb.html"_;"title=",c">,[b,c+__a,c.html"_;"title=",c.html"_;"title=",[b,c">,[b,c+__a,c">,c.html"_;"title=",[b,c">,[b,c+__a,cb">,c">,[b,c+__a,c.html"_;"title=",c.html"_;"title=",[b,c">,[b,c+__a,c">,c.html"_;"title=",[b,c">,[b,c+__a,cb_\,_

In_other_words,_right_multiplication_by_any_element_''c''_is_a_derivation_(abstract_algebra).html" "title=",c">,[b,c+__a,cb.html" ;"title=",c">,[b,c+__a,c.html" ;"title=",c.html" ;"title=",[b,c">,[b,c+ a,c">,c.html" ;"title=",[b,c">,[b,c+ a,cb">,c">,[b,c+__a,c.html" ;"title=",c.html" ;"title=",[b,c">,[b,c+ a,c">,c.html" ;"title=",[b,c">,[b,c+ a,cb \,

In other words, right multiplication by any element ''c'' is a derivation (abstract algebra)">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''.html" ;"title="'b'', ''c''.html" ;"title="'a'', [''b'', ''c''">'a'', [''b'', ''c'' + [''c'', [''a'', ''b''">'b'', ''c''.html" ;"title="'a'', [''b'', ''c''">'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 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''(''V'') , of any vector space ''V'' can be turned into a Loday algebra such that

: _1\otimes \cdots \otimes a_n,xa_1\otimes \cdots a_n\otimes x\quad \texta_1,\ldots, a_n,x\in V.

This is the free Loday algebra over ''V''.

Leibniz algebras were discovered in 1965 by A. Bloh, who called them D-algebras. They attracted interest after Jean-Louis Loday noticed that the classical Chevalley–Eilenberg boundary map 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 well-defined for any Leibniz algebra. The homology ''HL''(''L'') of this chain complex is known as Leibniz homology. If ''L'' is the Lie algebra of (infinite) matrices over an associative ''R''-algebra A then Leibniz homology
of ''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:

:$( a \circ b ) \circ c = a \circ (b \circ c) + a \circ (c \circ b) .$

Notes

References

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{{DEFAULTSORT:Leibniz Algebra
Lie algebras
Non-associative algebras