# 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:

Zinbiel algebras were introduced by Jean-Louis Loday (1995). The name was proposed by Jean-Michel Lemaire as being "opposite" to Leibniz algebra.[1]

The symmetrised product

is associative.

A Zinbiel algebra is the Koszul dual concept to a Leibniz algebra. The free Zinbiel algebra over *V* is the tensor algebra with product

## References

- Loday 2001, p. 45

- Dzhumadil'daev, A.S.; Tulenbaev, K.M. (2005). "Nilpotency of Zinbiel algebras".
*J. Dyn. Control Syst*.**11**(2): 195–213. - Ginzburg, Victor; Kapranov, Mikhail (1994). "Koszul duality for operads".
*Duke Mathematical Journal*.**76**: 203–273. arXiv:0709.1228. doi:10.1215/s0012-7094-94-07608-4. MR 1301191. - Loday, Jean-Louis (1995). "Cup-product for Leibniz cohomology and dual Leibniz algebras" (PDF).
*Math. Scand*.**77**(2): 189–196. - Loday, Jean-Louis (2001).
*Dialgebras and related operads*. Lecture Notes in Mathematics.**1763**. Springer Verlag. pp. 7–66. - Zinbiel, Guillaume W. (2012), "Encyclopedia of types of algebras 2010", in Guo, Li; Bai, Chengming; Loday, Jean-Louis (eds.),
*Operads and universal algebra*, Nankai Series in Pure, Applied Mathematics and Theoretical Physics,**9**, pp. 217–298, arXiv:1101.0267, Bibcode:2011arXiv1101.0267Z, ISBN 9789814365116

This article is issued from
Wikipedia.
The text is licensed under Creative
Commons - Attribution - Sharealike.
Additional terms may apply for the media files.