Gorenstein subrings of invariants under Hopf algebra actions

Kirkman, Ellen; Kuzmanovich, James; Zhang, J.J.

doi:10.1016/j.jalgebra.2009.08.018

Cited by 77 publications

(108 citation statements)

References 19 publications

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“…The latter algebras are called Artin-Schelter (AS) regular algebras of global dimension 2. Previous work [12], [14], [15], [16] demonstrates that there is a rich invariant theory in this context. The goal of this paper is to classify noncommutative analogues of linear actions of finite subgroups of SL 2 (k) on AS regular algebras of global dimension 2 and study the resulting rings of invariants.…”

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confidence: 87%

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Quantum binary polyhedral groups and their actions on quantum planes

Chan

Kirkman

Walton

et al. 2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We classify quantum analogues of actions of finite subgroups G of SL 2 (k) on commutative polynomial rings k [u, v]. More precisely, we produce a classification of pairs (H, R), where H is a finite dimensional Hopf algebra that acts inner faithfully and preserves the grading of an Artin-Schelter regular algebra R of global dimension two. Remarkably, the corresponding invariant rings R H share similar regularity and Gorenstein properties as the invariant rings k [u, v] G in the classical setting. We also present several questions and directions for expanding this work in noncommutative invariant theory.

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confidence: 87%

“…We use the convention introduced in [15,Section 3], in particular, we assume that H acts on A on the right. Let Y denote the local cohomology module H (c) Since hdet H A is trivial, we have that…”

Section: Hopf Actions With Trivial Homological Determinantmentioning

confidence: 99%

Quantum binary polyhedral groups and their actions on quantum planes

Chan

Kirkman

Walton

et al. 2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…This work contributes to the field of noncommutative invariant theory in the sense of studying quantum analogues of group actions on commutative k-algebras. Here, we restrict our attention to the actions of finite quantum groups, i.e., finite dimensional Hopf algebras, as these objects and their actions on (quantum) k-algebras have been the subject of recent research in noncommutative invariant theory, including [8], [10], [16], [18], [27], [29], [34], [35], [37]. The two important classes of finite dimensional Hopf algebras H are those that are semisimple (as a k-algebra) and those that are pointed (namely, all simple H-comodules are 1-dimensional).…”

Section: Introductionmentioning

confidence: 99%

“…(1) 27 Twists u q (gl n ) J + , u q (gl n ) J − for n ≥ 2 A n−1 ×A n−1 Prop. 29 Twists u q (sl n ) J + , u q (sl n ) J − for n ≥ 2 A n−1 ×A n−1 Cor.…”

Section: Introductionmentioning

confidence: 99%

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Pointed Hopf Actions on Fields, I

Etingof

Walton

2015

Transformation Groups

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Actions of semisimple Hopf algebras H over an algebraically closed field of characteristic zero on commutative domains were classified recently by the authors in [18]. The answer turns out to be very simple -if the action is inner faithful, then H has to be a group algebra. The present article contributes to the non-semisimple case, which is much more complicated. Namely, we study actions of finite dimensional (not necessarily semisimple) Hopf algebras on commutative domains, particularly when H is pointed of finite Cartan type.The work begins by reducing to the case where H acts inner faithfully on a field; such a Hopf algebra is referred to as Galois-theoretical. We present examples of such Hopf algebras, which include the Taft algebras, uq(sl 2 ), and some Drinfeld twists of other small quantum groups. We also give many examples of finite dimensional Hopf algebras which are not Galois-theoretical. Classification results on finite dimensional pointed Galois-theoretical Hopf algebras of finite Cartan type will be provided in the sequel, Part II, of this study.

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