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Fundamentalnaya i Prikladnaya Matematika, 2005, Volume 11, Issue 3, Pages 57–78 (Mi fpm828)  

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
References:
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 $q^2\ne1$.
English version:
Journal of Mathematical Sciences (New York), 2007, Volume 144, Issue 2, Pages 3909–3925
DOI: https://doi.org/10.1007/s10958-007-0244-9
Bibliographic databases:
UDC: 512.552
Language: Russian
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://mathscinet.ams.org/mathscinet-getitem?mr=2176680}
\zmath{https://zbmath.org/?q=an:1119.17001}
\elib{https://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{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-34250219574}
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  • https://www.mathnet.ru/eng/fpm/v11/i3/p57
  • This publication is cited in the following 25 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Фундаментальная и прикладная математика
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