Geodesic
Binary $(-1,1)$-bimodules over semisimple algebras
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1605-1625.

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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. 
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S. V. Pchelintsev. Binary $(-1,1)$-bimodules over semisimple algebras. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1605-1625. http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a24/

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