# 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**

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

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:

## Notes

- ↑ Barnes, Donald W. (July 2011). "Some Theorems on Leibniz Algebras".
*Communications in Algebra*.**39**(7): 2463–2472. doi:10.1080/00927872.2010.489529. - ↑ Patsourakos, Alexandros (26 November 2007). "On Nilpotent Properties of Leibniz Algebras".
*Communications in Algebra*.**35**(12): 3828–3834. doi:10.1080/00927870701509099. - ↑ 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. - ↑ 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.