Invariant theory of Artin-Schelter regular
                    algebras: a survey

Kirkman, Ellen

doi:10.1090/conm/673/13489

Cited by 13 publications

(6 citation statements)

References 30 publications

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“…Modern work in noncommutative invariant theory has focused on group and Hopf algebra actions on Artin-Schelter regular rings, which may be regarded as noncommutative versions of polynomial rings. We refer the interested reader to the survey of Kirkman for an overview of work in this area [70]. In this short section we review some invariant-theoretic results regarding GWAs.…”

Section: Invariant Theorymentioning

confidence: 99%

Fixed rings of generalized Weyl algebras

Gaddis

Won

2019

Journal of Algebra

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Generalized Weyl Algebras (GWAs) appear in diverse areas of mathematics including mathematical physics, noncommutative algebra, and representation theory. We study the invariants of quantum GWAs under finite automorphisms. We extend a theorem of Jordan and Wells and apply it to determine the fixed ring of quantum GWAs under diagonal automorphisms. We further study properties of the fixed rings, including global dimension, rigidity, and simplicity. IntroductionThroughout, k is a field and all algebras are associative k-algebras.The Shephard-Todd-Chevalley (STC) Theorem [6,24] gives conditions for the fixed ring of a polynomial ring by a finite group of linear automorphisms to again be a polynomial ring. More recently, there has been significant interest in studying STC-like theorems in noncommutative algebra, in particular whether the fixed ring of an (N-graded) Artin-Schelter regular algebra again has this property [15].To consider this problem outside of the N-graded setting, one could ask whether the fixed ring of a (twisted) Calabi-Yau algebra is again (twisted) Calabi-Yau. Algebras satisfying this property have attracted much interest of late [9,18,20,21]. Since polynomial rings are (trivially) Calabi-Yau, this is indeed a reasonable generalization.An important family of Z-graded (twisted) Calabi-Yau algebras are the generalized Weyl algebras (GWAs) and so they serve as a good test case of the STC question in this setting. Kirkman and Kuzmanovich [16] have proposed a version of the STC Theorem for GWAs, essentially asking when the fixed ring of a GWA again has GWA structure. We propose a strengthening of this: to determine when the fixed ring of a Calabi-Yau GWA is again a Calabi-Yau GWA.Generalized Weyl algebras were named by Bavula [2]. They have been studied extensively by many authors prior to and post Bavula's definition. Notably, the interested reader is directed to the work of Hodges [10, 11], Jordan [12], Joseph [14], Rosenberg [23], Smith [26], and Stafford [27]. There are severalimportant families of algebras that may be constructed as GWAs. In the classical case, this includes the Weyl algebras and primitive quotients of U (sl 2 ). We will primarily be concerned with a subclass known as quantum GWAs, which includes quantum planes, quantum Weyl algebras, and primitive quotients of U q (sl 2 ).The proposal of Kirkman and Kuzmanovich [16] mentioned above has its basis in the work of Jordan and Wells [13] and their study of fixed rings of GWAs by automorphisms that fix the base ring. Won and the first-named author showed that it is possible to diagonalize any filtered automorphism of a classical GWA 1 with quadratic defining polynomial so that the Jordan and Wells result may be applied [8]. The methods and results in this paper vary wildly from these aforementioned works.Results. First, in Section 2, we study automorphisms of quantum GWAs. We use the classification of their automorphism groups by Suárez-Alvarez and Vivas [28] in order to classify their finite subgroups (Proposition 2.6). In Section 3 w...

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Section: Invariant Theorymentioning

confidence: 99%

Fixed rings of generalized Weyl algebras

Gaddis

Won

2019

Journal of Algebra

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“…Let k be an algebraically closed field of characteristic 0. Symmetries of noncommutative algebras over k have been extensively studied in the context of Hopf algebra actions and coactions, with notable success for connected k-algebras (see, e.g., [Mon93,Kir16,CKWZ16,EW14]). However, extending these results to not-necessarily-connected k-algebras is not a straightforward task.…”

Section: Introductionmentioning

confidence: 99%

Symmetries of algebras captured by actions of weak Hopf algebras

Calderon,

Huang,

Wicks

et al. 2023

Preprint

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In this paper, we present a generalization of well-established results regarding symmetries of 𝕜-algebras, where 𝕜 is a field. Traditionally, for a 𝕜-algebra A, the group 𝕜-algebra automorphisms of A captures the symmetries of A via group actions. Similarly, the Lie algebra of derivations of A captures the symmetries of A via Lie algebra actions. In this paper, given a category C whose objects possess 𝕜-linear monoidal categories of modules, we introduce an object SymC (A) that captures the symmetries of A via actions of objects in C. Our study encompasses various categories whose objects include groupoids, Lie algebroids, and more generally, cocommutative weak Hopf algebras. Notably, we demonstrate that for a positively graded non-connected 𝕜-algebra A, some of its symmetries are naturally captured within the weak Hopf framework. 2020 Mathematics Subject Classification. 18B40, 16T05, 18M05.

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“…In recent years, there has been a strong interest in generalising results from commutative invariant theory, such as those comprising the Auslander-McKay correspondence, to a noncommutative setting; see [Kir16] for a survey of recent results. A common approach in noncommutative invariant theory is to replace the polynomial ring R by an Artin-Schelter (AS) regular algebra A, and the finite group G GL(n, k) by a semisimple Hopf algebra H. It is then possible to define an invariant ring A H , as well as an analogue of the skew group algebra, called the smash product, denoted A # H. One can then study properties of A H and A # H, with A H playing the role of the coordinate ring of a noncommutative (often singular) variety.…”

Section: Introductionmentioning

confidence: 99%

Superpotentials and Quiver Algebras for Semisimple Hopf Actions

Crawford¹

2021

Preprint

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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.

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Invariant theory of Artin-Schelter regular algebras: a survey

Cited by 13 publications

References 30 publications

Fixed rings of generalized Weyl algebras

Fixed rings of generalized Weyl algebras

Symmetries of algebras captured by actions of weak Hopf algebras

Superpotentials and Quiver Algebras for Semisimple Hopf Actions

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