Document Zbl 07896847

Binary \((-1,1)\)-bimodules over semisimple algebras. (Russian. English summary) Zbl 07896847

Sib. Èlektron. Mat. Izv. 20, No. 2, 1605-1625 (2023).

Summary: It is proved that the irreducible binary \((-1,1)\)-bimodule over simple algebra with a unit is alternative. A criterion for alternativeness (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 coordinatization theorem is proved for the matrix algebra \(\mathrm{M}_n(\Phi)\) 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 \(\Phi(t)\).

MSC:

17D20	\((\gamma, \delta)\)-rings, including \((1,-1)\)-rings
16D20	Bimodules in associative algebras

Keywords:

strictly \((-1, 1)\)-algebra; \((-1, 1)\)-algebra; binary \((-1, 1)\)-algebra; \(\mathfrak{M}\)-bimodule; irreducible bimodule; complete reducibility

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