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Alexey Sergeevich Gordienko

Publications and source records attributed to Alexey Sergeevich Gordienko.

11 recordsLinked to original sources
arXiv2015

Asymptotics of H-identities for associative algebras with an H-invariant radical

We prove the existence of the Hopf PI-exponent for finite dimensional associative algebras $A$ with a generalized Hopf action of an associative algebra $H$ with $1$ over an algebraically closed field of characteristic $0$ assuming only the invariance of the Jacobson radical $J(A)$ under the $H$-action and the existence of the decomposition of $A/J(A)$ into the sum of $H$-simple algebras. As a consequence, we show that the analog of Amitsur's conjecture holds for $G$-codimensions of finite dimensional associative algebras over a field of characteristic $0$ with an action of an arbitrary group $G$ by automorphisms and anti-automorphisms and for differential codimensions of finite dimensional associative algebras with an action of an arbitrary Lie algebra by derivations.

math.RA↗
arXiv2015

Semigroup graded algebras and codimension growth of graded polynomial identities

We show that if $T$ is any of four semigroups of two elements that are not groups, there exists a finite dimensional associative $T$-graded algebra over a field of characteristic $0$ such that the codimensions of its graded polynomial identities have a non-integer exponent of growth. In particular, we provide an example of a finite dimensional graded-simple semigroup graded algebra over an algebraically closed field of characteristic $0$ with a non-integer graded PI-exponent, which is strictly less than the dimension of the algebra. However, if $T$ is a left or right zero band and the $T$-graded algebra is unital, or $T$ is a cancellative semigroup, then the $T$-graded algebra satisfies the graded analog of Amitsur's conjecture, i.e. there exists an integer graded PI-exponent. Moreover, in the first case it turns out that the ordinary and the graded PI-exponents coincide. In addition, we consider related problems on the structure of semigroup graded algebras.

math.RA↗
arXiv2014

Algebras simple with respect to a Sweedler's algebra action

Algebras simple with respect to an action of Sweedler's algebra $H_4$ deliver the easiest example of $H$-module algebras that are $H$-simple but not necessarily semisimple. We describe finite dimensional $H_4$-simple algebras and prove the analog of Amitsur's conjecture for codimensions of their polynomial $H_4$-identities. In particular, we show that the Hopf PI-exponent of an $H_4$-simple algebra $A$ over an algebraically closed field of characteristic $0$ equals $\dim A$. The groups of automorphisms preserving the structure of an $H_4$-module algebra are studied as well.

math.RA↗
arXiv2014

Amitsur's conjecture for associative algebras with a generalized Hopf action

We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of several generalizations of polynomial identities for finite dimensional associative algebras over a field of characteristic 0, including G-identities for any finite (not necessarily Abelian) group G and H-identities for a finite dimensional semisimple Hopf algebra H. In addition, we prove that the Hopf PI-exponent of Sweedler's 4-dimensional algebra with the action of its dual equals 4.

math.RA↗
arXiv2014

Structure of H-(co)module Lie algebras

Let $L$ be a finite dimensional Lie algebra over a field of characteristic $0$. Then by the original Levi theorem, $L = B \oplus R$ where $R$ is the solvable radical and $B$ is some maximal semisimple subalgebra. We prove that if $L$ is an $H$-(co)module algebra for a finite dimensional (co)semisimple Hopf algebra $H$, then $R$ is $H$-(co)invariant and $B$ can be chosen to be $H$-(co)invariant too. Moreover, the nilpotent radical $N$ of $L$ is $H$-(co)invariant and there exists an $H$-sub(co)module $S\subseteq R$ such that $R=S\oplus N$ and $[B,S]=0$. In addition, the $H$-(co)invariant analog of the Weyl theorem is proved. In fact, under certain conditions, these results hold for an $H$-comodule Lie algebra $L$, even if $H$ is infinite dimensional. In particular, if $L$ is a Lie algebra graded by an arbitrary group $G$, then $B$ can be chosen to be graded, and if $L$ is a Lie algebra with a rational action of a reductive affine algebraic group $G$ by automorphisms, then $B$ can be chosen to be $G$-invariant. Also we prove that every finite dimensional semisimple $H$-(co)module Lie algebra over a field of characteristic $0$ is a direct sum of its minimal $H$-(co)invariant ideals.

math.RA↗
arXiv2014

On the formula for the PI-exponent of Lie algebras

We prove that one of the conditions in M.V. Zaicev's formula for the PI-exponent and in its natural generalization for the Hopf PI-exponent, can be weakened. Using the modification of the formula, we prove that if a finite dimensional semisimple Lie algebra acts by derivations on a finite dimensional Lie algebra over a field of characteristic $0$, then the differential PI-exponent coincides with the ordinary one. Analogously, the exponent of polynomial $G$-identities of a finite dimensional Lie algebra with a rational action of a connected reductive affine algebraic group $G$ by automorphisms, coincides with the ordinary PI-exponent. In addition, we provide a simple formula for the Hopf PI-exponent and prove the existence of the Hopf PI-exponent itself for $H$-module Lie algebras whose solvable radical is nilpotent, assuming only the $H$-invariance of the radical, i.e. under weaker assumptions on the $H$-action, than in the general case. As a consequence, we show that the analog of Amitsur's conjecture holds for $G$-codimensions of all finite dimensional Lie $G$-algebras whose solvable radical is nilpotent, for an arbitrary group $G$.

math.RA↗
arXiv2014

Algebras simple with respect to a Taft algebra action

Algebras simple with respect to an action of a Taft algebra $H_{m^2}(ζ)$ deliver an interesting example of $H$-module algebras that are $H$-simple but not necessarily semisimple. We describe finite dimensional $H_{m^2}(ζ)$-simple algebras and prove the analog of Amitsur's conjecture for codimensions of their polynomial $H_{m^2}(ζ)$-identities. In particular, we show that the Hopf PI-exponent of an $H_{m^2}(ζ)$-simple algebra $A$ over an algebraically closed field of characteristic $0$ equals $\dim A$. The groups of automorphisms preserving the structure of an $H_{m^2}(ζ)$-module algebra are studied as well.

math.RA↗
arXiv2013

Graded polynomial identities, group actions, and exponential growth of Lie algebras

Consider a finite dimensional Lie algebra L with an action of a finite group G over a field of characteristic 0. We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial G-identities of L. As a consequence, we prove the analog of Amitsur's conjecture for graded codimensions of any finite dimensional Lie algebra graded by a finite Abelian group.

math.RA↗
arXiv2013

Co-stability of radicals and its applications to PI-theory

We prove that if A is a finite dimensional associative H-comodule algebra over a field F for some involutory Hopf algebra H not necessarily finite dimensional, where either char F = 0 or char F > dim A, then the Jacobson radical J(A) is an H-subcomodule of A. In particular, if A is a finite dimensional associative algebra over such a field F, graded by any group, then the Jacobson radical J(A) is a graded ideal of A. Analogous results hold for nilpotent and solvable radicals of finite dimensional Lie algebras over a field of characteristic 0. We use the results obtained to prove the analog of Amitsur's conjecture for graded polynomial identities of finite dimensional associative algebras over a field of characteristic 0, graded by any group.

math.RA↗
arXiv2012

Amitsur's conjecture for polynomial H-identities of H-module Lie algebras

Consider a finite dimensional H-module Lie algebra L over a field of characteristic 0 where H is a Hopf algebra. We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial H-identities of L under some assumptions on H. In particular, the conjecture holds when H is finite dimensional semisimple. As a consequence, we obtain the analog of Amitsur's conjecture for graded codimensions of any finite dimensional Lie algebra graded by an arbitrary group and for G-codimensions of any finite dimensional Lie algebra with a rational action of a reductive affine algebraic group G by automorphisms and anti-automorphisms.

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