22 citations · 129 across the 18 of their papers we have counts for
23 papers · 1 filter
Hopf algebra polynomial identities in the Sweedler algebra
Alexey Gordienko, Maria Morozova
Using Mac Lane's PROPs, one can introduce multilinear polynomial identities that involve all the operations of a Hopf algebra: the (co)multiplication, the (co)unit and the antipode…
Graded group actions and generalized -actions compatible with gradings
A. S. Gordienko
We introduce the notion of a graded group action on a graded algebra or, which is the same, a group action by graded pseudoautomorphisms. An algebra with such an action is a natura…
(Co)module algebras and their generalizations
Alexey Gordienko
This manuscript is an extended version of the author's habilitation thesis defended on May 21, 2021 at M.V. Lomonosov Moscow State University. It is devoted to the study of (co)sta…
-universal Hopf algebras (co)acting on -algebras
Ana Agore, Alexey Gordienko, Joost Vercruysse
We develop a theory which unifies the universal (co)acting bi/Hopf algebras as studied by Sweedler, Manin and Tambara with the recently introduced \cite{AGV1} bi/Hopf-algebras that…
Equivalences of (co)module algebra structures over Hopf algebras
Ana Agore, Alexey Gordienko, Joost Vercruysse
We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) equivalence of gradings. We show that for eac…
Categories and weak equivalences of graded algebras
Alexey Gordienko, Ofir Schnabel
When one studies the structure (e.g. graded ideals, graded subspaces, radicals, ...) or graded polynomial identities of graded algebras, the grading group itself does not play an i…
Lie algebras simple with respect to a Taft algebra action
Alexey Gordienko
We classify finite dimensional -simple -module Lie algebras over an algebraically closed field of characteristic where is the th Taf…
On weak equivalences of gradings
Alexey Gordienko, Ofir Schnabel
When one studies the structure (e.g. graded ideals, graded subspaces, radicals, ...) or graded polynomial identities of graded algebras, the grading group itself does not play an i…
Semigroup graded algebras and graded PI-exponent
Alexey Gordienko, Geoffrey Janssens, Eric Jespers
Let be a finite semigroup and let be a finite dimensional -graded algebra. We investigate the exponential rate of growth of the sequence of graded codimensions $c_n^S(A)…
On -simple not necessarily associative algebras
Alexey Gordienko
An algebra A with a generalized H-action is a generalization of an H-module algebra where H is just an associative algebra with 1 and a relaxed compatibility condition between the…
Actions of Ore extensions and growth of polynomial -identities
Alexey Gordienko
We show that if is a finite dimensional associative -module algebra for an arbitrary Hopf algebra , then the proof of the analog of Amitsur's conjecture for -codimensi…
Semigroup graded algebras and codimension growth of graded polynomial identities
Alexey Sergeevich Gordienko
We show that if is any of four semigroups of two elements that are not groups, there exists a finite dimensional associative -graded algebra over a field of characteristic $…
Algebras simple with respect to a Taft algebra action
Alexey Sergeevich Gordienko
Algebras simple with respect to an action of a Taft algebra deliver an interesting example of -module algebras that are -simple but not necessarily semisimple. W…
Algebras simple with respect to a Sweedler's algebra action
Alexey Sergeevich Gordienko
Algebras simple with respect to an action of Sweedler's algebra deliver the easiest example of -module algebras that are -simple but not necessarily semisimple. We desc…
-modules and polynomial identities with integer coefficients
Alexey Gordienko, Geoffrey Janssens
We show that, like in the case of algebras over fields, the study of multilinear polynomial identities of unitary rings can be reduced to the study of proper polynomial identities.…
Co-stability of radicals and its applications to PI-theory
Alexey Sergeevich Gordienko
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…
Asymptotics of H-identities for associative algebras with an H-invariant radical
Alexey Sergeevich Gordienko
We prove the existence of the Hopf PI-exponent for finite dimensional associative algebras with a generalized Hopf action of an associative algebra with over an algebra…
On the formula for the PI-exponent of Lie algebras
Alexey Sergeevich Gordienko
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…
Derivations, gradings, actions of algebraic groups, and codimension growth of polynomial identities
A. S. Gordienko, M. V. Kochetov
Suppose a finite dimensional semisimple Lie algebra acts by derivations on a finite dimensional associative or Lie algebra over a field of characteristic . We…
Amitsur's conjecture for polynomial H-identities of H-module Lie algebras
Alexey Sergeevich Gordienko
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…
Amitsur's conjecture for associative algebras with a generalized Hopf action
Alexey Sergeevich Gordienko
We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of several generalizations of polynomial identities for finite dimensional associative algebras…
Graded polynomial identities, group actions, and exponential growth of Lie algebras
Alexey Sergeevich Gordienko
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…
Codimensions of polynomial identities of representations of Lie algebras
Alexey Sergeevich Gordienko
Consider a representation where is a Lie algebra and is a finite dimensional vector space. We prove the analog of Amitsur's conjecture on a…