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mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a (right) Leibniz algebra, named after
Gottfried Wilhelm Leibniz
Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to ...
, sometimes called a Loday algebra, after
Jean-Louis Loday
Jean-Louis Loday (12 January 1946 – 6 June 2012) was a French mathematician who worked on cyclic homology and who introduced Leibniz algebras (sometimes called Loday algebras) and Zinbiel algebras.
He occasionally used the pseudonym Guillaume W ...
, 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 (abstract algebra)">derivation
Derivation may refer to:
Language
* Morphological derivation, a word-formation process
* Parse tree or concrete syntax tree, representing a string's syntax in formal grammars
Law
* Derivative work, in copyright law
* Derivation proceeding, a ...
. 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 identity is equivalent to Jacobi's identity (
'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
In representation theory, a branch of mathematics, Engel's theorem states that a finite-dimensional Lie algebra \mathfrak g is a nilpotent Lie algebra if and only if for each X \in \mathfrak g, the adjoint map
:\operatorname(X)\colon \mathfrak \ ...
still holds for Leibniz algebras and that a weaker version of the
Levi–Malcev 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 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 the Leibniz homology of ''L'' is the tensor algebra over the
Hochschild homology
In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a fiel ...
of ''A''.
A ''
Zinbiel algebra In mathematics, a Zinbiel algebra or dual Leibniz algebra is a module over a commutative ring with a bilinear product satisfying the defining identity:
:(a \circ b) \circ c = a \circ (b \circ c) + a \circ (c \circ b).
Zinbiel algebras were intro ...
'' is the
Koszul dual concept to a Leibniz algebra. It has as defining identity:
:
Notes
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{{DEFAULTSORT:Leibniz Algebra
Lie algebras
Non-associative algebras