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Fundam. Prikl. Mat., 2005, Volume 11, Issue 3, Pages 57–78 (Mi fpm828)  

This article is cited in 2 scientific papers (total in 3 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 $q^2\ne1$.

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English version:
Journal of Mathematical Sciences (New York), 2007, 144:2, 3909–3925

Bibliographic databases:

UDC: 512.552

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

Citation in format AMSBIB
\Bibitem{Dzh05}
\by A.~S.~Dzhumadil'daev
\paper Zinbiel algebras under $q$-commutator
\jour Fundam. Prikl. Mat.
\yr 2005
\vol 11
\issue 3
\pages 57--78
\mathnet{http://mi.mathnet.ru/fpm828}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2176680}
\zmath{https://zbmath.org/?q=an:1119.17001}
\elib{http://elibrary.ru/item.asp?id=9027763}
\transl
\jour J. Math. Sci.
\yr 2007
\vol 144
\issue 2
\pages 3909--3925
\crossref{https://doi.org/10.1007/s10958-007-0244-9}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-34250219574}


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    This publication is cited in the following articles:
    1. Dzhumadil'daev A.S., “Anti-commutative algebras with skew-symmetric identities”, J. Algebra Appl., 8:2 (2009), 157–180  crossref  mathscinet  zmath  isi
    2. L. A. Bokut', E. I. Zelmanov, P. Zusmanovich, V. G. Kac, L. G. Makar-Limanov, Yu. I. Manin, S. P. Novikov, A. N. Parshin, V. P. Platonov, I. A. Taimanov, U. U. Umirbaev, I. P. Shestakov, “Askar Serkulovich Dzhumadil'daev (on his 60th birthday)”, Russian Math. Surveys, 72:4 (2017), 777–781  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. Dzhumadil'daev A.S. Ismailov N.A., “Polynomial Identities of Bicommutative Algebras, Lie and Jordan Elements”, Commun. Algebr., 46:12 (2018), 5241–5251  crossref  mathscinet  zmath  isi  scopus
  • Фундаментальная и прикладная математика
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