Identities on Two-Dimensional Algebras
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Identities on Two-Dimensional Algebras H. Ahmed1* , U. Bekbaev2** , and I. Rakhimov3, 4*** (Submitted by S. N. Tronin) 1 2
Department of Mathematics, Faculty of Science, Taiz University, Taiz, Yemen
Department of Mathematical and Natural Sciences, Turin Polytechnic University in Tashkent (TTPU), Tashkent, Uzbekistan 3
4
Department of Mathematics, Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA (UiTM), Shah Alam, Malaysia
V. I. Romanovski Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan Received November 19, 2019; revised February 26, 2020; accepted March 3, 2020
Abstract—In the paper we provide some polynomial identities for finite-dimensional algebras. A list of well known single polynomial identities is exposed and the classification of all two-dimensional algebras with respect to these identities is given. DOI: 10.1134/S1995080220090024 Keywords and phrases: algebra, polynomial identity, structure constants.
1. INTRODUCTION It is known that many important algebras in use are so called PI-algebras, that is, algebras satisfying a certain set of polynomial identities. Therefore, the classification of such algebras, up to isomorphism, is of a great interest. Earlier we have given classification results for some important classes of two-dimensional PI-algebras [2–5, 7]. Subalgebras, idempotents, ideals and quasi-units of two- dimensional algebras are described in [1]. In this paper we consider a list of some important polynomial identities which have appeared earlier in the theory of algebras and present a classification of two-dimensional algebras with respect to these identities. For other results related to the classification problem and the problems raised in this paper we refer the reader to [8–14]. The organization of the paper is as follows. In the next section we introduce definitions, notations and results needed in the course of the study followed by two section where we present main results of the paper. In Section 3 we provide some polynomial identities for finite-dimensional algebras. The last section is devoted to the classification of two-dimensional algebras with respect to the identities specified. 2. PRELIMINARIES In this paper an algebra (A, ·) means a vector space A over a field F with a given bilinear map · : A × A → A, (u, v) → u · v and often we drop · in the writing. *
E-mail: [email protected] E-mail: [email protected] *** E-mail: [email protected] **
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⎛ ⎞ α α α α 1 2 3 4⎠ If A such a two-dimensional algebra and e = (e1 , e2 ) is a fixed basis then by A = ⎝ β1 β2 β3 β4 we denote its matrix of structure constants (MSC) with respect to this basis, i.e., e1 e1 = α1 e1 + β1 e2 ,
e1 e2 = α2 e1 + β2 e2 ,
e2 e1 = α3 e1 + β3 e2 ,
e2 e2 = α4 e1 + β4 e2 .
Further it is assumed that the basis e is fixed and we do not make difference between an algebra A and its MSC A with respect to this basis. The classification problem of all two-dimensional algebras over any field F, where any second and third degree polynomial pos
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