Kemer's theory for H-module algebras with application to the PI exponent

Varieties of Null-Filiform Leibniz Algebras Under the Action of Hopf Algebras

“…In the general case of an H -algebra only partial results are known about the existence of such exponent. If H is finite dimensional and semisimple acting on an associative algebra over a field of characteristic 0, then Karasik proved in [35] the H -exponent exists and is an integer. It is easy to see Taft's algebras are not semisimple algebras.…”

The Crossed Product of Finite Hopf C*-Algebra and C*-Algebra

“…Hopf algebra actions are used in certain contexts in order to achieve some important results, i.e., Kemer's theory for H-module algebras, applying to the polynomial identity exponent [7]. What is more, to reveal the quantum symmetry in quantum spin models, finite Hopf C * -algebra actions are used to construct the quantum double.…”

Actions of Ore extensions and growth of polynomialH-identities

“…That led to the proof of the analog of Amitsur's conjecture for finite dimensional associative algebras graded by any group not necessarily finite [22]. In 2015 Yaakov Karasik [38] proved the existence of integer PIexp H (A) for (not necessarily finite dimensional) H-module PI-algebras A for finite dimensional semisimple Hopf algebras H.…”

“…As we have mentioned above, the proofs in all previous papers [1,2,17,21,23,24,38] worked only in the case when the Jacobson radical J(A) was an H-submodule or A was H-simple itself. In the current article we do not assume that the Jacobson radical of A is H-invariant, replacing the Wedderburn -Mal'cev theorem by its weak analog (Lemma 2.6) which still makes it possible to transfer the computations to H-simple algebras.…”