Geodesic
Zinbiel algebras under $q$-commutator
Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 3, pp. 57-78.

Voir la notice de l'article provenant de la source Math-Net.Ru

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 $q^2\ne1$. 
@article{FPM_2005_11_3_a3,
     author = {A. S. Dzhumadil'daev},
     title = {Zinbiel algebras under $q$-commutator},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {57--78},
     publisher = {mathdoc},
     volume = {11},
     number = {3},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2005_11_3_a3/}
}
TY  - JOUR
AU  - A. S. Dzhumadil'daev
TI  - Zinbiel algebras under $q$-commutator
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2005
SP  - 57
EP  - 78
VL  - 11
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2005_11_3_a3/
LA  - ru
ID  - FPM_2005_11_3_a3
ER  - 
%0 Journal Article
%A A. S. Dzhumadil'daev
%T Zinbiel algebras under $q$-commutator
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2005
%P 57-78
%V 11
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2005_11_3_a3/
%G ru
%F FPM_2005_11_3_a3
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/

[1] Agrachev A., Gamkrelidze R., “Eksponentsialnoe predstavlenie potokov i khronologicheskoe ischislenie”, Mat. sb., 107:4 (1978), 487–532 | MR

[2] Agrachev A., Gamkrelidze R., “Khronologicheskie algebry i nestatsionarnye vektornye polya”, Itogi nauki i tekhn. Ser. Probl. geometrii, 11, VINITI, M., 1980, 135–176 | MR

[3] Zhevlakov K. A., Slinko A. M., Shestakov I. P., Shirshov A. I., Koltsa, blizkie k assotsiativnym, Nauka, M., 1976

[4] Kavskii M., “Khronologicheskie algebry: kombinatorika i upravlenie”, Itogi nauki i tekhn., 64, VINITI, M., 1999, 144–178 | MR

[5] Dzhumadil'daev A. S., “Novikov–Jordan algebras”, Comm. Algebra, 30:11 (2002), 5205–5238 | MR

[6] Loday J.-L., “Cup-product for Leibniz cohomology and dual Leibniz algebras”, Math. Scand., 77:2 (1995), 189–196 | MR | Zbl