Rigidity of graded regular algebras

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

doi:10.1090/s0002-9947-08-04571-6

Cited by 40 publications

(66 citation statements)

References 32 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: 86%

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: 86%

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 means that (1 − t) n−2 T r A R (g| A R , t) is analytic at t = 1 and g| A R cannot be a quasi-reflection. Since (A R ) G/R is regular, G/R must contain a quasi-reflection by [KKZ1,Theorem 2.4]. Hence G/R is trivial and G = R.…”

Section: Definitionsmentioning

confidence: 97%

“…In [KKZ1] the authors assume that k is algebraically closed. For simplicity and our convenience we continue to assume that k is algebraically closed since we will use several results from [KKZ1], though all the main assertions in this paper hold without that assumption. The opposite ring of an algebra A is denoted by A op .…”

Section: Definitionsmentioning

confidence: 99%

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Shephard–Todd–Chevalley Theorem for Skew Polynomial Rings

Kirkman

Kuzmanovich

Zhang

2009

Algebr Represent Theor

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Abstract. We prove the following generalization of the classical ShephardTodd-Chevalley Theorem. Let G be a finite group of graded algebra automorphisms of a skew polynomial ring A := kp ij [x 1 , · · · , xn]. Then the fixed subring A G has finite global dimension if and only if G is generated by quasireflections. In this case the fixed subring A G is isomorphic a skew polynomial ring with possibly different p ij 's. A version of the theorem is proved also for abelian groups acting on general quantum polynomial rings.

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“…In this paper we use our work in noncommutative invariant theory to propose several notions of a noncommutative graded complete intersection. Moreover, the existence of noncommutative analogues of commutative complete intersection invariant subalgebras broadens our continuing project of establishing an invariant theory for finite groups acting on Artin-Schelter regular algebras that is parallel to classical invariant theory (see [28][29][30][31][32]). …”

Section: Introductionmentioning

confidence: 99%