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

where the sum is over all shuffles.[1]


  1. 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
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