Searcharxiv⌕ Search

arXiv subjects

I. Rakhimov

Publications and source records attributed to I. Rakhimov.

10 recordsLinked to original sources
arXiv2026

On three-dimensional associative algebras

This paper addresses the classification problem of associative algebras over arbitrary base fields. We present a list of three-dimensional associative algebras with canonical representatives of the isomorphism classes for fields of characteristic different from two and three. We also compare our lists with the most recent classifications over the complex numbers and with the nilpotent case over arbitrary base fields in dimension three, adding some comments.

math.RA↗
arXiv2024

Classification of Frobenius algebra structures on two-dimensional vector space over any base field

Classifying Frobenius algebras is a key question that has been addressed in various contexts. The structure of finite-dimensional Frobenius algebras depends on the base field and the dimension of the algebra, leading to different classification results depending on whether the base field has characteristic zero, characteristic $p$, or other properties. Frobenius algebras over fields of characteristic zero have been well-studied, often related to semisimple algebra theory. The behavior of Frobenius algebras over fields of positive characteristic presents new challenges, with connections to modular representation theory. In the paper, we first classify all associative algebra structures on a two-dimensional vector space over any base field equipped with a non-degenerate bilinear form. We then identify which of these are Frobenius algebras. Lists of canonical representatives of the isomorphism classes of these algebras are provided for a base field with characteristics not equal to two or three, as well as for characteristics two and three.

math.RA↗
arXiv2024

A Complete classification of two-dimensional endo-commutative algebras over an arbitrary field

In the paper, we consider the class of so-called endo-commutative algebras. From the identity imposed to specify this class, one can easily see that the product in this class preserves the square of elements. We give a complete classification of this class, up to isomorphism, over any basic field in dimension two. This elementary and self-contained exposition uses the recent result of one of the authors on description of the category of all $2$-dimensional algebras over any basic field.

math.RA↗
arXiv2019

On identities of $2$-dimensional algebras

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 $2$-dimensional algebras with respect to these identities is given.

math.RA↗
arXiv2019

Subalgebras, idempotents, ideals and quasi-units of two-dimensional algebras

All subalgebras, idempotents, left(right) ideals and left quasi-units of two-dimensional algebras are described. Classification of algebras with given number of subalgebras, left(right) ideals are provided. In particular, a list of isomorphism classes of all simple two-dimensional algebras is given. In the study of ideals and subalgebras the number of them depend on roots of certain system of polynomials at structure constants of the algebra. We also give explicit forms of the polynomials.

math.RA↗