Narrow Results

The quantum Schrödinger algebra is a quantized universal enveloping algebra of the (centrally extended) Schrödinger algebra. It can be realized as the smash product of the quantum group $U_q({𝔰𝔩}_2)$ and its a module algebra. For generic $q$, in this paper we study the BGG category ${𝒪}$ for the quantum Schrödinger algebra. Then we use the result to determine its center, which is a polynomial ring in two variables.

In this paper, we use semigroupoids to describe a notion of algebraic bundles, mostly motivated by Fell ($C^{\ast }$-algebraic) bundles, and the sectional algebras associated to them. As the main motivational example, Steinberg algebras may be regarded as the sectional algebras of trivial (direct product) bundles. Several theorems which relate geometric and algebraic constructions — via the construction of a sectional algebra — are widely generalized: Direct products bundles by semigroupoids correspond to tensor products of algebras; semidirect products of bundles correspond to “naïve” crossed products of algebras; skew products of graded bundles correspond to smash products of graded algebras; Quotient bundles correspond to quotient algebras. Moreover, most of the results hold in the non-Hausdorff setting. In the course of this work, we generalize the definition of smash products to groupoid graded algebras. As an application, we prove that whenever $𝜃$ is a ∧-preaction of a discrete inverse semigroupoid $S$ on an ample (possibly non-Hausdorff) groupoid ${𝒢}$, the Steinberg algebra of the associated groupoid of germs is naturally isomorphic to a crossed product of the Steinberg algebra of ${𝒢}$ by $S$. This is a far-reaching generalization of analogous results which had been proven in particular cases.

This is a continuation of a previous study initiated by the third author on nonlocal vertex bialgebras and smash product nonlocal vertex algebras. In this paper, we study a notion of right $H$-comodule nonlocal vertex algebra for a nonlocal vertex bialgebra $H$ and give a construction of deformations of vertex algebras with a right $H$-comodule nonlocal vertex algebra structure and a compatible $H$-module nonlocal vertex algebra structure. We also give a construction of $\varphi$-coordinated quasi modules for smash product nonlocal vertex algebras. As an example, we give a family of quantum vertex algebras by deforming the vertex algebras associated to non-degenerate even lattices.

Steenrod's problem of realizing modules has been solved by Arnold and Vogel for groups of a cyclic nature, and has counterexamples given by Carlsson, Vogel, Benson and Habegger, most concretely for Z/2×Z/2. The last two authors develop nontrivial methods to realize other modules for this group, but they do not extend to the general case Z/p×Z/p. In this paper we give some positive results for groups (Z/p)^m. Of interest we find that relevant sequences and their duals are both realizable. Also we establish an invariance of the Steenrod problem suggested by Benson and Habeggers' approach.

Let be a Hopf algebroid, and A a left -module algebra. This paper is concerned with the smash product algebra A#H over Hopf algebroids. In this paper, we investigate separable extensions for module algebras over Hopf algebroids. As an application, we obtain a Maschke-type theorem for A#H-modules over Hopf algebroids, which generalizes the corresponding result given by Cohen and Fischman in [Hopf algebra actions, J. Algebra100 (1986) 363–379]. Furthermore, based on the work of Kadison and Szlachányi in [Bialgebroid actions on depth two extensions and duality, Adv. Math.179 (2003) 75–121], we construct a Morita context connecting A#H and the invariant subalgebra of on A.

In this paper, we introduce the concept of partial groupoid actions on $R$-semicategories as well as we give criteria for existence of a globalization of it. This point of view is a generalization of the notions of partial groupoid actions on rings and partial group action on an $R$-semicategory. We also define the notions of partial skew groupoid category, smash product and describe functorial relations between them, in particular we show that the smash product is a Galois covering of its associated skew groupoid category.

In this work, we deal with partial (co)actions of multiplier Hopf algebras on not necessarily unital algebras. Our main goal is to construct a Morita context relating the coinvariant algebra $R^{\underline{coA}}$ with a certain subalgebra of the smash product $R\#Â$. Besides that, we present the notion of partial Galois coaction, which is closely related to this Morita context.

In this paper, some properties of vertex algebras acted by a Hopf algebra will be given. We prove that $V\#H$ is an ${𝒮}$-local vertex algebra, where $V$ is an $H$-module vertex algebra. In addition, we extend the ${𝒮}$-locality to the quantum vertex algebras.