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

• Kosmann-Schwarzbach, Yvette (1996). "From Poisson algebras to Gerstenhaber algebras". Annales de l'Institut Fourier. 46 (5): 1243–1274. doi:10.5802/aif.1547.
• Loday, Jean-Louis (1993). "Une version non commutative des algèbres de Lie: les algèbres de Leibniz". Enseign. Math. (2). 39 (3–4): 269–293.
• Loday, Jean-Louis & Teimuraz, Pirashvili (1993). "Universal enveloping algebras of Leibniz algebras and (co)homology". Mathematische Annalen. 296 (1): 139–158. doi:10.1007/BF01445099.
• Bloh, A. (1965). "On a generalization of the concept of Lie algebra". Dokl. Akad. Nauk SSSR. 165: 471–473.
• Bloh, A. (1967). "Cartan-Eilenberg homology theory for a generalized class of Lie algebras". Dokl. Akad. Nauk SSSR. 175 (8): 824–826.
• Dzhumadil'daev, A.S.; Tulenbaev, K.M. (2005). "Nilpotency of Zinbiel algebras". J. Dyn. Control Syst. 11 (2): 195–213.
• Ginzburg, V.; Kapranov, M. (1994). "Koszul duality for operads". Duke Math. J. 76: 203–273. arXiv:0709.1228. doi:10.1215/s0012-7094-94-07608-4.