Zinbiel algebra
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In mathematics, a Zinbiel algebra or dual Leibniz algebra is a module over a commutative ring with a bilinear product satisfying the defining identity:
- [math]\displaystyle{ (a \circ b) \circ c = a \circ (b \circ c) + a \circ (c \circ b). }[/math]
Zinbiel algebras were introduced by Jean-Louis Loday (1995). The name was proposed by Jean-Michel Lemaire as being "opposite" to Leibniz algebra.[1]
In any Zinbiel algebra, the symmetrised product
- [math]\displaystyle{ a \star b = a \circ b + b \circ a }[/math]
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
- [math]\displaystyle{ (x_0 \otimes \cdots \otimes x_p) \circ (x_{p+1} \otimes \cdots \otimes x_{p+q}) = x_0 \sum_{(p,q)} (x_1,\ldots,x_{p+q}), }[/math]
where the sum is over all [math]\displaystyle{ (p,q) }[/math] shuffles.[1]
References
- ↑ 1.0 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. doi:10.1215/s0012-7094-94-07608-4.
- Loday, Jean-Louis (1995). "Cup-product for Leibniz cohomology and dual Leibniz algebras". Math. Scand. 77 (2): 189–196. http://www.math.uiuc.edu/K-theory/0015/cup_product.pdf.
- Loday, Jean-Louis (2001). Dialgebras and related operads. Lecture Notes in Mathematics. 1763. Springer Verlag. pp. 7–66. http://www.math.uiuc.edu/K-theory/0333/.
- Zinbiel, Guillaume W. (2012), "Encyclopedia of types of algebras 2010", in Guo, Li; Bai, Chengming; Loday, Jean-Louis, Operads and universal algebra, Nankai Series in Pure, Applied Mathematics and Theoretical Physics, 9, pp. 217–298, ISBN 9789814365116, Bibcode: 2011arXiv1101.0267Z, http://www.worldscibooks.com/mathematics/8222.html
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