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Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports], 2023, Volume 20, Issue 2, Pages 1605–1625
DOI: https://doi.org/doi.org/10.33048/semi.2023.20.099
(Mi semr1662)
 

Mathematical logic, algebra and number theory

Binary $(-1,1)$-bimodules over semisimple algebras

S. V. Pchelintsev

Department of Mathematics, Finance University under the Government of the Russian Federation, Leningradsky prospect 49, 125993, Moscow, Russia
References:
Abstract: It is proved that the irreducible binary $(-1,1)$-bimodule over simple algebra with a unit is alternative. A criterion for alterna-tiveness (hence, complete reducibility) of unital binary $(-1,1)$-bimodule over a semisimple finite-dimensional algebra is obtained. It is proved that every unital strictly $(-1,1)$-bimodule over a finite-dimensional semisimple associative and commutative algebra is associative. The coordinateization theorem is proved for the matrix algebra ${\rm M}_n(\Phi)$ of order $n\geq 3$ in the class of binary $(-1,1)$-algebras. Finally, the following examples of indecomposable $(-1,1)$-bimodules are constructed: the non-unital bimodule over $1$-dimensional algebra $\Phi e$; the unital bimodule over a $2$-dimensional composition algebra $\Phi e_1 \oplus \Phi e_2$; the unital $(-1,1)$-bimodule over a quadratic extension $\Phi(\sqrt{\lambda})$ of the ground field; the unital strictly $(-1,1)$-bimodule over the field of fractionally rational functions of one variable $\Phi(t)$.
Keywords: strictly $(-1,1)$-algebra, $(-1,1)$-algebra, binary $(-1,1)$-algebra, ${\mathfrak M}$-bimodule, irreducible bimodule, complete reducibility.
Funding agency Grant number
Fundação de Amparo à Pesquisa do Estado de São Paulo 2023/01159_5
Received September 12, 2023, published December 29, 2023
Document Type: Article
UDC: 512.554.5
MSC: 17A70, 17D15
Language: Russian
Citation: S. V. Pchelintsev, “Binary $(-1,1)$-bimodules over semisimple algebras”, Sib. Èlektron. Mat. Izv., 20:2 (2023), 1605–1625
Citation in format AMSBIB
\Bibitem{Pch23}
\by S.~V.~Pchelintsev
\paper Binary $(-1,1)$-bimodules over semisimple algebras
\jour Sib. \`Elektron. Mat. Izv.
\yr 2023
\vol 20
\issue 2
\pages 1605--1625
\mathnet{http://mi.mathnet.ru/semr1662}
\crossref{https://doi.org/doi.org/10.33048/semi.2023.20.099}
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