The kS3-Module Algebra Structures on M3(k)

Let S 3 be the symmetric group on three elements. Let k be a field and M 3 ( k ) be the full matrix algebra of 3 × 3 -matrices over k. In this paper, the k S 3 -module algebra structures on M 3 ( k ) are described, and classified up to isomorphism.

Factorizable Module Algebras

The aim of this paper is to introduce and study a large class of $\mathfrak{g}$-module algebras that we call factorizable by generalizing the Gauss factorization of square or rectangular matrices. This class includes coordinate algebras of corresponding reductive groups G, their parabolic subgroups, basic affine spaces, and many others. It turns out that products of factorizable algebras are also factorizable and it is easy to create a factorizable algebra out of virtually any $\mathfrak{g}$-module algebra. We also have quantum versions of all these constructions in the category of $U_{q}(\mathfrak{g})$-module algebras. Quite surprisingly, our quantum factorizable algebras are naturally acted on by the quantized enveloping algebra $U_{q}(\mathfrak{g}^{\ast })$ of the dual Lie bialgebra $\mathfrak{g}^{\ast }$ of $\mathfrak{g}$.

On the Semiprimitivity and Semiprimality Problems for Partial Smash Products

In this paper we discuss about the semiprimitivity and the semiprimality of partial smash products. Let H be a semisimple Hopf algebra over a field 𝕜 and let A be a left partial H-module algebra. We study the H-prime and the H-Jacobson radicals of A and their relations with the prime and the Jacobson radicals of A#H, respectively. In particular, we prove that if A is H-semiprimitive, then A#H is semiprimitive provided that all irreducible representations of A are finite-dimensional, or A is an affine PI-algebra over 𝕜 and 𝕜 is a perfect field, or A is locally finite. Moreover, we prove that A#H is semiprime provided that A is an H-semiprime PI-algebra, generalizing to the setting of partial actions the known results for global actions of Hopf algebras.

Inner actions of weak Hopf algebras

Let [Formula: see text] be an associative ring and [Formula: see text] idempotent elements of [Formula: see text]. In this paper we introduce the notion of [Formula: see text]-invertibility for an element of [Formula: see text] and use it to define inner actions of weak Hopf algebras. Given a weak Hopf algebra [Formula: see text] and an algebra [Formula: see text], we present sufficient conditions for [Formula: see text] to admit an inner action of [Formula: see text]. We also prove that if [Formula: see text] is a left [Formula: see text]-module algebra then [Formula: see text] acts innerly on the smash product [Formula: see text] if and only if [Formula: see text] is a quantum commutative weak Hopf algebra.

An extended form of Majid’s double biproduct

Let [Formula: see text] be a bialgebra. Let [Formula: see text] be a linear map, where [Formula: see text] is a left [Formula: see text]-module algebra, and a coalgebra with a left [Formula: see text]-weak coaction. Let [Formula: see text] be a linear map, where [Formula: see text] is a right [Formula: see text]-module algebra, and a coalgebra with a right [Formula: see text]-weak coaction. In this paper, we extend the construction of two-sided smash coproduct to two-sided crossed coproduct [Formula: see text]. Then we derive the necessary and sufficient conditions for two-sided smash product algebra [Formula: see text] and [Formula: see text] to be a bialgebra, which generalizes the Majid’s double biproduct in [Double-bosonization of braided groups and the construction of [Formula: see text], Math. Proc. Camb. Philos. Soc. 125(1) (1999) 151–192] and the Wang–Wang–Yao’s crossed coproduct in [Hopf algebra structure over crossed coproducts, Southeast Asian Bull. Math. 24(1) (2000) 105–113].

A duality theorem for weak multiplier Hopf algebra actions

The main purpose of this paper is to unify the theory of actions of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras to one of actions of weak multiplier Hopf algebras introduced by Van Daele and Wang. Using such developed actions, we will define the notion of a module algebra over weak multiplier Hopf algebras and construct their smash products. The main result is the duality theorem for actions and their dual actions on the smash product of weak multiplier Hopf algebras. As an application, we recover the main results found in the literature for weak Hopf algebras, multiplier Hopf algebras and groupoids.

ON GORENSTEINNESS OF HOPF MODULE ALGEBRAS

Let H be a Hopf algebra with a bijective antipode, A an H-simple H-module algebra finitely generated as an algebra over the ground field and module-finite over its centre. The main result states that A has finite injective dimension and is, moreover, Artin–Schelter Gorenstein under the additional assumption that each H-orbit in the space of maximal ideals of A is dense with respect to the Zariski topology. Further conclusions are derived in the cases when the maximal spectrum of A is a single H-orbit or contains an open dense H-orbit.