On the Structure of Alternative Bimodules over Semisimple Artinian Algebras
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the Structure of Alternative Bimodules over Semisimple Artinian Algebras L. R. Borisova1* and
S. V. Pchelintsev1**
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Finance University under the Government, 49 Leningradsky Ave., Moscow, 125993 Russia Received October 8, 2019; revised October 8, 2019; accepted December 18, 2019
Abstract—The alternative bimodules over semisimple artinian algebras are studied. A bimodule is called almost reducible if it is a direct sum of an associative subbimodule and a completely reducible subbimodule. It is proved that if a semisimple algebra cannot be homomorphically mapped onto an associative division algebra, then an alternative bimodule above it is almost reducible. An example of an alternative bimodule over a field of rational functions of two variables, which is not almost reducible, is given. DOI: 10.3103/S1066369X20080010 Key words: alternative algebra, irreducible bimodule, almost reducible bimodule.
INTRODUCTION The notion of a bimodule (birepresentation) for an arbitrary class of algebras was introduced by S. Eilenberg [1]. Together with associative algebras and Lie algebras, the theory of birepresentations had been developed also for another classes of algebras. For alternative algebras, the main results were obtained by R.D. Schafer [2] and N. Jacobson [3], for Jordan algebras by N. Svartholm [4] and N. Jacobson [3], for Maltsev algebras by R. Carlsson [5] and E.N. Kuzmin [6]. For the classes of algebras above, there are theorems on complete reducibility of bimodules over semisimple finite-dimensional algebras, the structure of irreducibility of bimodules is described. I.P. Shestakov [7] proved complete reducibility of alternative bimodules over the algebra of generalized quaternions without restrictions on the dimension and the characteristic. Irreducible right alternative bimodules over the algebra Φ2 of matrices of the 2nd order were studied by L.I. Murakami and I.P. Shestakov [8]. It was proved that, over the algebra Φ2 , there exist infinitely many non-isomorphic irreducible right alternative bimodules, and not every right alternative bimodule over the algebra Φ2 is completely reducible. In [9], irreducible binary (−1, 1)-bimodules over simple finite-dimensional algebras were studied. In particular, it was proved that any irreducible binary (−1, 1)-bimodule over an algebra A is alternative in any of the following cases: A is a compositional algebra; A is a finite-dimensional simple alternative algebra with characteristic 0. In [10], irreducible alternative bimodules over simple alternative algebras of an arbitrary dimension were studied. A bimodule over an algebra A is called almost reducible, if it is a direct sum of an associative A-subbimodule and a completely reducible A-subbimodule. The objective of the present paper is the study of alternative bimodules over semisimple artinian algebras. The paper consists of the introduction and three sections. Section 1 overviews the known results. In section 2, we prove * **
E-mail: [email protected] E-mail: [email protected]
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BORISOVA, PCHELINTSEV
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