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,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.

In this sense, Leibniz algebras can be seen as a non-commutative generalization of Lie algebras, and the investigation of which theorems and properties of Lie algebras are still valid for Leibniz algebras is a recurrent theme in the literature.[1] For instance, it has been shown that Engel's theorem still holds for Leibniz algebras[2][3] and that a weaker version of Levi-Malcev theorem also holds.[4]

The tensor module, T(V) , of any vector space V can be turned into a Loday algebra such that

 [a_1\otimes \cdots \otimes a_n,x]=a_1\otimes \cdots a_n\otimes x\quad \text{for }a_1,\ldots, a_n,x\in V.

This is the free Loday algebra over V.

Leibniz algebras were discovered by A. Bloh in 1965 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

  1. Barnes, Donald W. (July 2011). "Some Theorems on Leibniz Algebras". Communications in Algebra 39 (7): 2463–2472. doi:10.1080/00927872.2010.489529.
  2. Patsourakos, Alexandros (26 November 2007). "On Nilpotent Properties of Leibniz Algebras". Communications in Algebra 35 (12): 3828–3834. doi:10.1080/00927870701509099.
  3. Sh. A. Ayupov; B. A. Omirov (1998). "On Leibniz Algebras". In Khakimdjanov, Y.; Goze, M.; Ayupov, Sh. Algebra and Operator Theory Proceedings of the Colloquium in Tashkent, 1997. Dordrecht: Springer. pp. 1–13. ISBN 9789401150729.
  4. Barnes, Donald W. (30 November 2011). "On Levi's theorem for Leibniz algebras". Bulletin of the Australian Mathematical Society 86 (02): 184–185. doi:10.1017/s0004972711002954.

References

This article is issued from Wikipedia - version of the Sunday, October 25, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.