Математическая логика, алгебра и теория чисел
Бинарно $(-1,1)$-бимодули над полупростыми алгебрами
С. В. Пчелинцев Department of Mathematics, Finance University under the Government of the Russian Federation, Leningradsky prospect 49, 125993, Moscow, Russia
Аннотация:
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
$\Phi(t)$.
Ключевые слова:
strictly
$(-1,1)$-algebra,
$(-1,1)$-algebra, binary
$(-1,1)$-algebra,
${\mathfrak M}$-bimodule, irreducible bimodule, complete reducibility.
УДК:
512.554.5
MSC: 17A70,
17D15 Поступила 12 сентября 2023 г., опубликована
29 декабря 2023 г.
DOI:
doi.org/10.33048/semi.2023.20.099