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Fundamentalnaya i Prikladnaya Matematika, 2005, Volume 11, Issue 3, Pages 57–78
(Mi fpm828)
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This article is cited in 24 scientific papers (total in 25 papers)
Zinbiel algebras under $q$-commutator
A. S. Dzhumadil'daev Kazakh-British Technical University
Abstract:
An algebra with the identity $t_1(t_2t_3)=(t_1t_2+t_2t_1)t_3$ is called Zinbiel. For example, $\mathbb C[x]$ under multiplication $a\circ b=b\int\limits_0^xa\,dx $ is Zinbiel. Let $a\circ_q b=a\circ b+q\,b\circ a$ be a $q$-commutator, where $q\in\mathbb C$. We prove that for any Zinbiel algebra $A$ the corresponding algebra under commutator $A^{(-1)}=(A,\circ_{-1})$ satisfies the identities $t_1t_2=-t_2t_1$ and $(t_1t_2)(t_3t_4)+(t_1t_4)(t_3t_2)=
\operatorname{jac}(t_1,t_2,t_3)t_4+\operatorname{jac}(t_1,t_4,t_3)t_2$, where $\operatorname{jac}(t_1,t_2,t_3)=(t_1t_2)t_3+(t_2t_3)t_1+(t_3t_1)t_2$. We find basic identities for $q$-Zinbiel algebras and prove that they form varieties equivalent to the variety of Zinbiel algebras if $q^2\ne1$.
Citation:
A. S. Dzhumadil'daev, “Zinbiel algebras under $q$-commutator”, Fundam. Prikl. Mat., 11:3 (2005), 57–78; J. Math. Sci., 144:2 (2007), 3909–3925
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https://www.mathnet.ru/eng/fpm828 https://www.mathnet.ru/eng/fpm/v11/i3/p57
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