Identities of graded algebras and codimension growth

Vercruysse

2021

J. Noncommut. Geom.

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 each equivalence class of (co)module algebra structures on a given algebra A, there exists a unique universal Hopf algebra H together with an H-(co)module structure on A such that any other equivalent (co)module algebra structure on A factors through the action of H. We study support equivalence and the universal Hopf algebras mentioned above for group gradings, Hopf-Galois extensions, actions of algebraic groups and cocommutative Hopf algebras. We show how the notion of support equivalence can be used to reduce the classification problem of Hopf algebra (co)actions. In particular, we apply support equivalence to study the asymptotic behaviour of codimensions of H-identities of a certain class of H-module algebras. This result proves the analogue (formulated by Yu. A. Bahturin) of Amitsur's conjecture which was originally concerned with ordinary polynomial identities.

Section: Equivalences Of Group Gradings and Group Actionsmentioning

confidence: 99%

Equivalences of (co)module algebra structures over Hopf algebras

Agore

Vercruysse

2021

J. Noncommut. Geom.

“…Remark. Conversely, for every matrix P ∈ M k (F ) such that P 2 = αE k for some α ∈ F , we can define the structure of an H 4 -simple algebra on M k (F ) ⊕ M k (F ) by (2), which is even Z 2 -simple.…”

Section: Semisimple H 4 -Simple Algebrasmentioning

confidence: 99%

Algebras simple with respect to a Sweedler's algebra action

2014

J. Algebra Appl.

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.Comment: 12 pages; minor misprints have been corrected; one reference has been adde

“…Zaicev [10,Theorem 6.5.2] for all associative PI-algebras. Alongside with ordinary polynomial identities of algebras, graded, differential, G-and H-identities are important too [4,5,6]. Usually, to find such identities is easier than to find the ordinary ones.…”

Section: Introductionmentioning

confidence: 99%

Co-stability of Radicals and Its Applications to PI-Theory

2016

Algebra Colloq.

Abstract. 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.