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Yuri Bahturin

Publications and source records attributed to Yuri Bahturin.

At least 19 recordsLinked to original sources
arXiv2026

Quantum regularity of finite dimensional semisimple algebras

In this paper, we establish a necessary and sufficient criterion for a finite-dimensional semisimple algebra over an algebraically closed field of characteristic $0$ to admit a regular quantum commutative decomposition. We apply this characterization to group algebras and show that a finite group algebra admits such a decomposition if and only if the underlying group is a peak group, that is, a finite group whose irreducible character degrees have a greatest element with respect to the divisibility ordering. We investigate the structure of peak groups and establish several criteria for their solvability in terms of the prime divisors of their largest irreducible character degree. In particular, we show that several important classes of groups are peak groups, while supersolvability alone does not imply the peak property. Finally, we construct a non-solvable peak group within the class of Frobenius groups.

math.RA↗
arXiv2026

Finite basis property for finite graded algebras

Let $G$ be a finite group and let $\F$ be a finite field. We prove that any finite-dimensional $G$-graded associative algebra $A$ over $\F$ has a finite basis for its $G$-graded polynomial identities.

math.RA↗
arXiv2026

On Regular Quantum Commutative Algebras

Let $K$ be an algebraically closed field of characteristic different from $2$. We provide a positive solution to the Bahturin--Regev conjecture in the general finite-dimensional (non-graded) setting, assuming that $\operatorname{char}(K)$ does not divide the quantum length of a minimal regular quantum commutative decomposition. Furthermore, we obtain a criterion, formulated in terms of regular quantum commutative decompositions, under which a set-grading on a semisimple associative algebra is realized as a group grading.

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arXiv2026

Locally finite varieties of nonassociative algebras

We study locally finite varieties (=primitive classes) of linear algebras over finite fields. We do not assume that our algebras are associative or Lie. We are interested in the basic properties of finite algebras in these varieties such as: nilpotence, solvability, simplicity, freeness, projectivity, and injectivity. We are also interested in the numerical estimates of the ratio of the number of algebras with various classical properties to the total number of all algebras of a fixed dimension $n$. Among these properties are having no proper nontrivial subalgebras or no nontrivial automorphisms, etc.

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arXiv2024

Superpolynomial identities of finite-dimensional simple algebras

We investigate the Grassmann envelope (of finite rank) of a finite-dimensional $\mathbb{Z}_2$-graded algebra. As a result, we describe the polynomial identities of $G_1(\mathcal{A})$, where $G_1$ stands for the Grassmann algebra with $1$ generator, and $\mathcal{A}$ is a $\mathbb{Z}_2$-graded-simple associative algebra. We also classify the conditions under which two associative $\mathbb{Z}_2$-graded-simple algebras share the same set of superpolynomial identities, i.e., the polynomial identities of its Grassmann envelope (in particular, of finite rank). Moreover, we extend the construction of the Grassmann envelope for the context of $Ω$-algebras and prove some of its properties. Lastly, we give a description of $\mathbb{Z}_2$-graded-simple $Ω$-algebras.

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arXiv2024

On generic properties of nilpotent algebras

We study general nilpotent algebras. The results obtained are new even for the classical algebras, such as associative or Lie algebras. We single out certain generic properties of finite-dimensional algebras, mostly over infinite fields. The notion of being generic in the class of $n$-generated algebras of an arbitrary primitive class of $c$-nilpotent algebras appears naturally in the following way. On the set of the isomorphism classes of such algebras one can introduce the structure of an algebraic variety. As a result, the subsets are endowed with the dimensions as algebraic varieties. A subset $Y$ of a set $X$ of lesser dimension can be viewed as negligible in $X$. For example, if $n\gg c$, we determine that an automorphism group of a generic algebra $P$ consists of the automorphisms, which are scalar modulo $P^2$. Generic ideals are in $I(P)$, the annihilator of $P$. In the case of classical nilpotent algebras as above, the generic algebras are graded by the degrees with respect to some generating sets.

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arXiv2022

Nilpotent algebras, implicit function theorem, and polynomial quasigroups

We study finite-dimensional nonassociative algebras. We prove the implicit function theorem for such algebras. This allows us to establish a correspondence between such algebras and quasigroups, in the spirit of classical correspondence between divisible torsion-free nilpotent groups and rational nilpotent Lie algebras. We study the related questions of the commensurators of nilpotent groups, filiform Lie algebras of maximal solvability length and partially ordered algebras.

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arXiv2021

Delta sets and polynomial identities in pointed Hopf algebras

We survey a vast array of known results and techniques in the area of polynomial identities in pointed Hopf algebras. Some new results are proven in the setting of Hopf algebras that appeared in papers of D. Radford and N. Andruskiewitsch - H.-J. Schneider.

math.QA↗
arXiv2019

Graded-division algebras over arbitrary fields

A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a classification of finite-dimensional graded-central graded-division algebras over an arbitrary field $\mathbb{F}$ can be reduced to the following three classifications, for each finite Galois extension $\mathbb{L}$ of $\mathbb{F}$: (1) finite-dimensional central division algebras over $\mathbb{L}$, up to isomorphism; (2) twisted group algebras of finite groups over $\mathbb{L}$, up to graded-isomorphism; (3) $\mathbb{F}$-forms of certain graded matrix algebras with coefficients in $Δ\otimes_{\mathbb{L}}\mathcal{C}$ where $Δ$ is as in (1) and $\mathcal{C}$ is as in (2). As an application, we classify, up to graded-isomorphism, the finite-dimensional graded-division algebras over the field of real numbers (or any real closed field) with an abelian grading group. We also discuss group gradings on fields.

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arXiv2019

Graded torsion-free ${\mathfrak{sl}_2(\mathbb{C})}$-modules of rank 2

In this paper we explore the possibility of endowing simple infinite-dimensional ${\mathfrak{sl}_2(\mathbb{C})}$-modules by the structure of the graded module. The gradings on finite-dimensional simple module over simple Lie algebras has been studied in [arXiv:1308.6089] and [arXiv:1601.03008].

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arXiv2018

Graded polynomial identities as identities of universal algebras

Let $A$ and $B$ be finite-dimensional simple algebras with arbitrary signature over an algebraically closed field. Suppose $A$ and $B$ are graded by a semigroup $S$ so that the graded identitical relations of $A$ are the same as those of $B$. Then $A$ is isomorphic to $B$ as an $S$-graded algebra.

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arXiv2018

On nonassociative graded-simple algebras over the field of real numbers

We extend the loop algebra construction for algebras graded by abelian groups to study graded-simple algebras over the field of real numbers (or any real closed field). As an application, we classify up to isomorphism the graded-simple alternative (nonassociative) algebras and graded-simple finite-dimensional Jordan algebras of degree 2. We also classify the graded-division alternative (nonassociative) algebras up to equivalence.

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arXiv2016

Graded identities of simple real graded division algebras

Let A and B be finite dimensional simple real algebras with division gradings by an abelian group G. In this paper we give necessary and sufficient conditions for the coincidence of the graded identities of A and B. We also prove that every finite dimensional simple real algebra with a G-grading satisfies the same graded identities as a matrix algebra over an algebra D with a division grading that is either a regular grading or a non-regular Pauli grading. Moreover we determine when the graded identities of two such algebras coincide. For graded simple algebras over an algebraically closed field it is known that two algebras satisfy the same graded identities if and only if they are isomorphic as graded algebras.

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