Hopf Actions Research Articles

Morita equivalence and globalization for partial Hopf actions on nonunital algebras

In this work, we investigate partial actions of a Hopf algebra [Formula: see text] on nonunital algebras and the associated partial smash products, with the objective of providing a framework where one may obtain results for both [Formula: see text]-algebras with local units and [Formula: see text]-categories. We show that our partial actions correspond to nonunital algebras in the category of partial representations of [Formula: see text]. The central problem of existence of a globalization for a partial action is studied in detail, and we provide sufficient conditions for the existence (and uniqueness) of a minimal globalization for associative algebras in general. Extending previous results by Abadie, Dokuchaev, Exel and Simon, we define Morita equivalence for partial Hopf actions, and we show that if two symmetrical partial actions are Morita equivalent then their standard globalizations are also Morita equivalent. Particularizing to the case of a partial action on an algebra with local units, we obtain several strong results on equivalences of categories of modules of partial smash products of algebras and partial smash products of [Formula: see text]-categories.

Hopf action on vertex algebras

In this paper, some properties of vertex algebras acted by a Hopf algebra will be given. We prove that [Formula: see text] is an [Formula: see text]-local vertex algebra, where [Formula: see text] is an [Formula: see text]-module vertex algebra. In addition, we extend the [Formula: see text]-locality to the quantum vertex algebras.

Superpotentials and Quiver Algebras for Semisimple Hopf Actions

We consider the action of a semisimple Hopf algebra H on an m-Koszul Artin–Schelter regular algebra A. Such an algebra A is a derivation-quotient algebra for some twisted superpotential w, and we show that the homological determinant of the action of H on A can be easily calculated using w. Using this, we show that the smash product A#H is also a derivation-quotient algebra, and use this to explicitly determine a quiver algebra Λ to which A#H is Morita equivalent, generalising a result of Bocklandt–Schedler–Wemyss. We also show how Λ can be used to determine whether the Auslander map is an isomorphism. We compute a number of examples, and show how several results for the quantum Kleinian singularities studied by Chan–Kirkman–Walton–Zhang follow using our techniques.

Pointed Hopf actions on quantum generalized Weyl algebras

We study actions of pointed Hopf algebras in the Z-graded setting. Our main result classifies inner-faithful actions of generalized Taft algebras on quantum generalized Weyl algebras which respect the Z-grading. We also show that generically the invariant rings of Taft actions on quantum generalized Weyl algebras are commutative rings.

Degree bounds for Hopf actions on Artin–Schelter regular algebras

We study semisimple Hopf algebra actions on Artin–Schelter regular algebras and prove several upper bounds on the degrees of the minimal generators of the invariant subring, and on the degrees of syzygies of modules over the invariant subring. These results are analogues of results for group actions on commutative polynomial rings proved by Noether, Fogarty, Fleischmann, Derksen, Sidman, Chardin, and Symonds.

POINTED HOPF ACTIONS ON CENTRAL SIMPLE DIVISION ALGEBRAS

We examine actions of finite-dimensional pointed Hopf algebras on central simple division algebras in characteristic 0. (By a Hopf action we mean a Hopf module algebra structure.) In all examples considered, we show that the given Hopf algebra does admit a faithful action on a central simple division algebra, and we construct such a division algebra. This is in contrast to earlier work of Etingof and Walton, in which it was shown that most pointed Hopf algebras do not admit faithful actions on fields. We consider all bosonizations of Nichols algebras of finite Cartan type, small quantum groups, generalized Taft algebras with non-nilpotent skew primitive generators, and an example of non-Cartan type.

Induced Hopf Galois structures and their local Hopf Galois modules

The regular subgroup determining an induced Hopf Galois structure for a Galois extension $L/K$ is obtained as the direct product of the corresponding regular groups of the inducing subextensions. We describe here the associated Hopf algebra and Hopf action of an induced structure and we prove that they are obtained by tensoring the corresponding inducing objects. In order to deal with their associated orders we develop a general method to compute bases and free generators in terms of matrices coming from representation theory of Hopf modules. In the case of an induced Hopf Galois structure it allows us to decompose the associated order, assuming that inducing subextensions are arithmetically disjoint.

Hopf actions of some quantum groups on path algebras

Our first collection of results parametrize (filtered) actions of a quantum Borel Uq(b)⊂Uq(sl2) on the path algebra of an arbitrary (finite) quiver. When q is a root of unity, we give necessary and sufficient conditions for these actions to factor through corresponding finite-dimensional quotients, generalized Taft algebras T(r,n) and small quantum groups uq(sl2).In the second part of the paper, we shift to the language of tensor categories. Here we consider a quiver path algebra equipped with an action of a Hopf algebra H to be a tensor algebra in the tensor category of representations H. Such a tensor algebra is generated by an algebra and bimodule in this tensor category. Our second collection of results describe the corresponding bimodule categories via an equivalence with categories of representations of certain explicitly described quivers with relations.

Computation of Hopf Galois structures on separable extensions and classification of those for degree twice an odd prime power

A Hopf Galois structure on a finite field extension [Formula: see text] is a pair [Formula: see text], where [Formula: see text] is a finite cocommutative [Formula: see text]-Hopf algebra and [Formula: see text] a Hopf action. In this paper, we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of a given degree and several properties of those. We show a table which summarizes the program results. Besides, for separable field extensions of degree [Formula: see text], with [Formula: see text] an odd prime number, we prove that the occurrence of some type of Hopf Galois structure may either imply or exclude the occurrence of some other type. In particular, for separable field extensions of degree [Formula: see text], we determine exactly the possible sets of Hopf Galois structure types.

Computation of Hopf Galois structures on low degree separable extensions and classification of those for degrees $p^2$ and $2p$

A Hopf Galois structure on a finite field extension L/K is a pair (H,µ), where H is a finite cocommutative K-Hopf algebra and µ a Hopf action. In this paper we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of degree up to eleven and several properties of those. Besides, we exhibit several results on Hopf Galois structures inspired by the program output. We prove that if (H,µ) is an almost classically Hopf Galois structure, then it is the unique Hopf Galois structure with underlying Hopf algebra H up to isomorphism. For p an odd prime, we prove that a separable extension of degree p2 may have only one type of Hopf Galois structure and determine those of cyclic type; we determine as well the Hopf Galois structures on separable extensions of degree 2p. We highlight the richness of the results obtained for extensions of degree 8 by computing an explicit example and presenting some tables which summarize these results.

Multiplier Hopf algebras: Globalization for partial actions

In partial action theory, a pertinent question is whenever given a partial action of a Hopf algebra [Formula: see text] on an algebra [Formula: see text], it is possible to construct an enveloping action. The authors Alves and Batista, in [M. Alves and E. Batista, Globalization theorems for partial Hopf (co)actions and some of their applications, groups, algebra and applications, Contemp. Math. 537 (2011) 13–30], have shown that this is always possible if [Formula: see text] is unital. We are interested in investigating the situation, where both algebras [Formula: see text] and [Formula: see text] are not necessarily unitary. A nonunitary natural extension for the concept of Hopf algebras was proposed by Van Daele, in [A. Van Daele, Multiplier Hopf algebras, Trans. Am. Math. Soc. 342 (1994) 917–932], which is called multiplier Hopf algebra. Therefore, we will consider partial actions of multipliers Hopf algebras on algebras with a nondegenerate product and we will present a globalization theorem for this structure. Moreover, Dockuchaev et al. in [Globalizations of partial actions on nonunital rings, Proc. Am. Math. Soc. 135 (2007) 343–352], have shown when group partial actions on nonunitary algebras are globalizable. Based on this paper, we will establish a bijection between globalizable group partial actions and partial actions of a multiplier Hopf algebra.

Pertinency of Hopf actions and quotient categories of Cohen–Macaulay algebras

We study invariants and quotient categories of fixed subrings of Artin–Schelter regular algebras under Hopf algebra actions.

Gauge Enhancement of Super M-Branes Via Parametrized Stable Homotopy Theory

A key open problem in M-theory is the mechanism of "gauge enhancement", which supposedly makes M-branes exhibit the nonabelian gauge degrees of freedom that are seen perturbatively in the limit of 10d string theory. In fact, since only the twisted K-theory classes represented by nonabelian Chan-Paton gauge fields on D-branes have invariant meaning, the problem is really the lift to M-theory of the twisted K-theory classification of D-brane charges. Here we show how this problem has a solution by universal constructions in super homotopy theory, at least rationally. We recall how double dimensional reduction of super M-brane charges is described by the cyclification adjunction applied to the 4-sphere, and how M-theory degrees of freedom hidden at ADE-singularities are induced by the suspended Hopf action on the 4-sphere. Combining these, we demonstrate, at the level of rational homotopy theory, that gauge enhancement in M-theory is exhibited by lifting against the fiberwise stabilization of the unit of this cyclification adjunction on the A-type orbispace of the 4-sphere. This explains how the fundamental D6 and D8 brane cocycles can be lifted from twisted K-theory to a cohomology theory for M-brane charge, at least rationally.

Hopf actions on vertex operator algebras, II: Smash product

The smash product V#H of a Hopf algebra H and an H-module vertex operator algebra V are investigated. An(V)-theory and contragredient module theory are founded for V#H. If H is the group algebra of a finite subgroup of Aut(V), all the irreducible inequivalent admissible V#H-modules are classified. We also prove the Maschke theorem in vertex operator algebra version. Finally, we prove weak duality theorems for actions and coactions in vertex operator algebras version.

McKay correspondence for semisimple Hopf actions on regular graded algebras. II

We continue our study of the McKay correspondence for grading preserving actions of semisimple Hopf algebras H on (noncommutative) Artin–Schelter regular algebras A . Here, we establish correspondences between module categories over A^H , over A\#H , and over End _{A^H}(A) . We also study homological properties of (endomorphism rings of) maximal Cohen–Macaulay modules over A^H .

Braided Hochschild cohomology and Hopf actions

We showt hat the braided Hochschild cohomology, of an algebra in a suitably algebraic braided monoidal category, admits a graded ring structure under which it is braided commutative. We then give a canonical identification between the usual Hochschild cohomology ring of a smash product and the (derived) invariants of its braided Hochschild cohomology ring. We apply our results to identify the associative formal deformation theory of a smash product with its formal deformation theory as a module algebra over the given Hopf algebra (when the Hopf algebra is sufficiently semisimple). As a second application we deduce some structural results for the usual Hochschild cohomology of a smash product, and discuss specific implications for finite group actions on smooth affine schemes.

Quasimodular Hecke algebras and Hopf actions

Let \Gamma=\Gamma(N) be a principal congruence subgroup of SL_2(\mathbb Z) . In this paper, we extend the theory of modular Hecke algebras due to Connes and Moscovici to define the algebra \mathcal Q(\Gamma) of quasimodular Hecke operators of level \Gamma . Then, \mathcal Q(\Gamma) carries an action of “the Hopf algebra \mathcal H_1 of codimension 1 foliations” that also acts on the modular Hecke algebra \mathcal A(\Gamma) of Connes and Moscovici. However, in the case of quasimodular forms, we have several new operators acting on the quasimodular Hecke algebra \mathcal Q(\Gamma) . Further, for each \sigma\in SL_2(\mathbb Z) , we introduce the collection \mathcal Q_\sigma(\Gamma) of quasimodular Hecke operators of level \Gamma twisted by \sigma . Then, \mathcal Q_\sigma(\Gamma) is a right \mathcal Q(\Gamma) -module and is endowed with a pairing (\_\_,\_\_):\mathcal Q_\sigma(\Gamma)\otimes \mathcal Q_\sigma(\Gamma)\longrightarrow \mathcal Q_\sigma(\Gamma). We show that there is a "Hopf action" of a certain Hopf algebra \mathfrak{h}_1 on the pairing on \mathcal Q_\sigma(\Gamma) . Finally, for any \sigma\in SL_2(\mathbb Z) , we consider operators acting between the levels of the graded module \mathbb Q_\sigma(\Gamma)=\underset{m\in \mathbb Z}{\oplus}\mathcal Q_{\sigma(m)}(\Gamma) , where \sigma(m)=\begin{pmatrix} 1 & m \\ 0 & 1 \\ \end{pmatrix}\cdot \sigma for any m\in \mathbb Z . The pairing on \mathcal Q_\sigma(\Gamma) can be extended to a graded pairing on \mathbb Q_\sigma(\Gamma) and we show that there is a Hopf action of a larger Hopf algebra \mathfrak{h}_{\mathbb Z}\supseteq \mathfrak{h}_1 on the pairing on \mathbb Q_\sigma(\Gamma) .

Hopf actions on vertex operator algebras

The Hopf actions on vertex operator algebras are investigated. If the action is semisimple, a Schur–Weyl type decomposition is obtained. When the Hopf algebra is finite dimensional and the action is faithful, the action is a group action. Moreover if the Hopf algebra is finite dimensional and the action is semisimple and inner faithful, the action is also a group action. In this case, inner faithfulness is equivalent to faithfulness.

McKay correspondence for semisimple Hopf actions on regular graded algebras, I

In establishing a more general version of the McKay correspondence, we prove Auslander's theorem for actions of semisimple Hopf algebras H on noncommutative Artin–Schelter regular algebras A of global dimension two, where A is a graded H-module algebra, and the Hopf action on A is inner faithful with trivial homological determinant. We also show that each fixed ring AH under such an action arises as an analogue of a coordinate ring of a Kleinian singularity.

Modular Hecke algebras over Möbius categories

We extend the modular Hecke operators of Connes and Moscovici by taking values in the modular incidence algebra M [ C ] over a Möbius category C . Here M is the ‘modular tower’ consisting of all modular forms of all weights across all levels. We construct multiple product structures on the collection A ( Γ ) [ C ] of modular Hecke operators of level Γ over C , where Γ is a principal congruence subgroup. These product structures are then shown to be well behaved with respect to Hopf actions on A ( Γ ) [ C ] . While A ( Γ ) [ C ] already carries an action of the Hopf algebra H 1 of codimension 1-foliations, the noncommutativity of the modular incidence algebra M [ C ] allows us to construct additional operators on A ( Γ ) [ C ] . The use of Möbius categories provides a single framework for describing modular Hecke operators taking values in various rings: from formal power series rings over M to arithmetic functions over M and algebras of upper triangular matrices with entries in M . Moreover, we use functors between Möbius categories to study relations between various modular Hecke algebras.