Invariant Theory of Finite Group Actions on Down-Up Algebras

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

doi:10.1007/s00031-014-9279-4

Cited by 16 publications

(15 citation statements)

References 30 publications

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“…Let R be a down-up algebra A(α, β) in the situation of Lemma 6.3, where β = −1 if α = 2. By Lemma 6.3, this algebra is generated by x and y and having a normal element Ω := xy − ayx such that g(Ω) = det(g)Ω for all g ∈ G. Consider R as an ungraded algebra and define a filtration F by setting F n R = (k ⊕ kx ⊕ ky ⊕ kΩ) n ⊆ R for all n ≥ 0 [KKZ1,Lemma 7.2(2)]. Then F n R is G-stable, consequently, A := gr F R is a connected graded algebra with G-action.…”

Section: Down-up Algebrasmentioning

confidence: 99%

Noncommutative Auslander theorem

Bao

Zhang

2018

Trans. Amer. Math. Soc.

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In 1960s Maurice Auslander proved the following important result. Let R be the commutative polynomial ring C[x 1 , . . . , xn], and let G be a finite small subgroup of GLn(C) acting on R naturally. Let A be the fixed subringThen the endomorphism ring of the right A-module R A is naturally isomorphic to the skew group algebra R * G. In this paper, a version of Auslander Theorem is proven for the following classes of noncommutative algebras: (a) noetherian PI local (or connected graded) algebras of finite injective dimension, (b) universal enveloping algebras of finite dimensional Lie algebras, and (c) noetherian graded down-up algebras.2000 Mathematics Subject Classification. Primary 16E65, 16E10.

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Section: Down-up Algebrasmentioning

confidence: 99%

Noncommutative Auslander theorem

Bao

Zhang

2018

Trans. Amer. Math. Soc.

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“…Among the results of [CKWZ2] is the following theorem. When A is AS regular of dimension 2 and H is a semisimple Hopf algebra acting on A with trivial homological determinant the invariant subalgebras A H (called "Kleinian quantum singularities") are all of the form C/ΩC for C a noetherian AS regular algebra of dimension 3 and Ω a normal regular element of C, and hence can be regarded as hypersurfaces in an AS regular algebra of dimension 3 (see [KKZ6,Theorem 0.1] and [CKWZ2]), and the explicit singularity Ω is given for each case. In [CKWZ2] it is shown further that the McKay quiver of H is isomorphic to the Gabriel quiver of the H-action on A, and the quivers that occur are Euclidean diagrams of types A, D, E, DL, and L.…”

Section: Artin-schelter Gorenstein Subrings Of Invariantsmentioning

confidence: 99%

Invariant theory of Artin-Schelter regular algebras: a survey

Kirkman¹

2016

Recent Developments in Representation Theory

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This is survey of results that extend notions of the classical invariant theory of linear actions by finite groups on k[x 1 , . . . , xn] to the setting of finite group or Hopf algebra H actions on an Artin-Schelter regular algebra A. We investigate when A H is AS regular, or AS Gorenstein, or a "complete intersection" in a sense that is defined. Directions of related research are explored briefly.

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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: 98%