Geodesic
Zinbiel algebras under $q$-commutator
Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 3, pp. 57-78
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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 $q^2\ne1$. 
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     author = {A. S. Dzhumadil'daev},
     title = {Zinbiel algebras under $q$-commutator},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2005_11_3_a3/}
}
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A. S. Dzhumadil'daev. Zinbiel algebras under $q$-commutator. Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 3, pp. 57-78. http://geodesic.mathdoc.fr/item/FPM_2005_11_3_a3/

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