Multiparameter twisted Weyl algebras

Futorny, Vyacheslav; Hartwig, Jonas T.

doi:10.1016/j.jalgebra.2011.11.004

Cited by 17 publications

(18 citation statements)

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“…The structure and representation theory of TGW algebras have been investigated in several papers. For example, families of simple weight modules were classified in [15], [14], [9], Whittaker modules classified in [7], bounded and unbounded * -representations studied in [16], generalized Serre relations were found in [10], and conditions for a TGW algebra to be a simple ring were given in [11].…”

Section: Introductionmentioning

confidence: 99%

“…Examples of TGW algebras include multiparameter quantized Weyl algebras [16], [9], [7], U(sl 2 ), U q (sl 2 ), Q i j -CCR (Canonical Commutation Relation) algebras [16], quantized Heisenberg algebras, extended OGZ algebras (r − 1, r, r + 1) [14], the Mickelsson-Zhelobenko algebra associated to the pair (gl n , gl n−1 ) [14], an example related to gl 3 [18], and examples attached to any symmetric generalized Cartan matrix [10].…”

Section: Introductionmentioning

confidence: 99%

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Twisted Generalized Weyl Algebras and Primitive Quotients of Enveloping Algebras

Hartwig

Serganova

2016

Algebr Represent Theor

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To each multiquiver Γ we attach a solution to the consistency equations associated to twisted generalized Weyl (TGW) algebras. This generalizes several previously obtained solutions in the literature. We show that the corresponding algebras (Γ) carry a canonical representation by differential operators and that (Γ) is universal among all TGW algebras with such a representation. We also find explicit conditions in terms of Γ for when this representation is faithful or locally surjective.By forgetting some of the structure of Γ one obtains a Dynkin diagram, D(Γ). We show that the generalized Cartan matrix of (Γ) coincides with the one corresponding to D(Γ) and that (Γ) contains graded homomorphic images of the enveloping algebra of the positive and negative part of the corresponding Kac-Moody algebra.Finally, we show that a primitive quotient U/J of the enveloping algebra of a finitedimensional simple Lie algebra over an algebraically closed field of characteristic zero is graded isomorphic to a TGW algebra if and only if J is the annihilator of a completely pointed (multiplicity-free) simple weight module. The infinite-dimensional primitive quotients in types A and C are closely related to (Γ) for specific Γ. We also prove one result in the affine case. 6 Appendix: Relation to a previous family of TGW algebras 36 Contents1 Preliminaries IntroductionTwisted generalized Weyl (TGW) algebras were introduced by Mazorchuk and Turowska in 1999 [15]. They are constructed from a commutative unital ring , an associative unital -algebra R, a collection of elements t = {t i } i∈I from the center of R, a collection of commutingautomorphisms σ = {σ i } i∈I of R, and a scalar matrix µ = (µ i j ) i, j∈I , by adjoining to R new non-commuting generators X i and Y i for i ∈ I, imposing some relations and taking the quotient by a certain radical (see Section 1.3 for the full definition). These algebras are denoted by µ (R, σ, t). They are naturally graded by the free abelian group on the index set I and come with a canonical map R → µ (R, σ, t) making them R-rings. In addition, if µ is symmetric, they can be equipped with an involution.The In a GWA one has X i X j = X j X i and Y i Y j = Y j Y i for all i, j. These relations need not hold in a general TGW algebra, where instead they are replaced by higher degree Serre-type relations [10].A question of particular importance is whether a given input datum (R, σ, t) actually gives rise to a non-trivial TGW algebra. Indeed, it can happen that the relations are contradictory so that µ (R, σ, t) = {0}, the algebra with one element [8, Ex. 2.8]. This does not occur for higher rank GWAs, as the conditions (1.1) implies the algebra is consistent. Thus it becomes important to find a substitute for (1.1) which ensures that a given TGW algebra is non-trivial. This question was solved in [8], in the case when the t i are not zero-divisors in R. The answer is the following. Theorem ([8]). Assume that the elements t i are not zero-divisors in R.Then the following set of equations is sufficient fo...

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Twisted Generalized Weyl Algebras and Primitive Quotients of Enveloping Algebras

Hartwig

Serganova

2016

Algebr Represent Theor

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“…This class of algebras was introduced by Bavula in [Ba1] and later investigated by various authors, see, in particular, [Ba2,BJ,DGO,Maz,Sh,Ha2] and references therein. Generalized Weyl algebras were further studied and generalized to natural higher rank analogues, see for example [BB,MT2,BO], and then to certain twisted and multi parameter versions, see for example [MT1,MT3,Ha1,FH1,FH2] and references therein. Many more papers studying generalized Weyl algebras can be found using Google search or searching MathSciNet.…”

Section: Introduction and Description Of The Resultsmentioning

confidence: 99%

Simple weight modules over weak generalized Weyl algebras

Lü

Mazorchuk

Zhao

2015

Journal of Pure and Applied Algebra

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In this paper we address the problem of classification of simple weight modules over weak generalized Weyl algebras of rank one. The principal difference between weak generalized Weyl algebras and generalized Weyl algebras is that weak generalized Weyl algebras are defined using an endomorphism rather than an automorphism of a commutative ring R. We reduce classification of simple weight modules over weak generalized Weyl algebras to description of the dynamics of the action of the above mentioned endomorphism on the set of maximal ideals. We also describe applications of our results to the study of generalized Heisenberg algebras.

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“…This class of algebras were also studied in [15], where rather than generalizing as we have suggested, the authors considered Benkart's algebras as members of a class of twisted generalized Weyl algebras. The class of algebras considered in [15] should be a fruitful place to apply the ideas and techniques used in our classification. For the algebras studied in this article, we believe that the rigidity of the relations means there are very few if any possibilities for locally nilpotent derivations.…”

Section: A Quantum Tame Generators Problemmentioning

confidence: 99%

On the automorphisms of quantum Weyl algebras

Kitchin

Launois

2019

Journal of Pure and Applied Algebra

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Motivated by Weyl algebra analogues of the Jacobian conjecture and the tame generators problem, we prove quantum versions of these problems for a family of analogues to the Weyl algebras. In particular, our results cover the Weyl-Hayashi algebras and tensor powers of a quantization of the first Weyl algebra which arises as a primitive factor algebra of U + q (so 5 ).Mathematics Subject Classification (2010). 16W35, 16S32, 16W20, 17B37.

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Multiparameter twisted Weyl algebras

Cited by 17 publications

References 19 publications

Twisted Generalized Weyl Algebras and Primitive Quotients of Enveloping Algebras

Twisted Generalized Weyl Algebras and Primitive Quotients of Enveloping Algebras

Simple weight modules over weak generalized Weyl algebras

On the automorphisms of quantum Weyl algebras

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