Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras

Hartwig, Jonas T.; Öinert, Johan

doi:10.1016/j.jalgebra.2012.10.009

Cited by 10 publications

(16 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This implies that if C and C ′ are two connected components of Γ, then [A g , A h ] = 0 for any g ∈ V C , h ∈ V C ′ . Thus, by [11,Thm. 5.3] and Lemma 4.7, it follows that C A (R E ) is commutative.…”

Section: Faithfulnessmentioning

confidence: 83%

“…The following theorem describes the rank of ker γ as a free abelian group, in terms of the graph Γ. In view of Section 6, it is parallel to [11,Thm. 5.8], since if Γ is a symmetric simple quiver, then every connected component is in equilibrium.…”

Section: Faithfulnessmentioning

confidence: 99%

“…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%

See 2 more Smart Citations

Twisted Generalized Weyl Algebras and Primitive Quotients of Enveloping Algebras

Hartwig

Serganova

2016

Algebr Represent Theor

View full text Add to dashboard Cite

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...

show abstract

Section: Faithfulnessmentioning

confidence: 83%

Section: Faithfulnessmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Twisted Generalized Weyl Algebras and Primitive Quotients of Enveloping Algebras

Hartwig

Serganova

2016

Algebr Represent Theor

View full text Add to dashboard Cite

show abstract

“…7.18] the following statements are proved. [11,Th. 7.18] in the more general context of so called R-finitistic TGW algebras.…”

Section: Centralizer Intersections and A Simplicity Criterion Let G mentioning

confidence: 99%

“…The form used in [16] requires the matrix µ to be symmetric (due to its formulation in terms of a certain involution on the TGWA). As is observed in [11], there is another way to define a bilinear form which works for general µ. It is given as follows.…”

Section: 4mentioning

confidence: 99%

Multiparameter twisted Weyl algebras

Futorny

Hartwig²

2012

Journal of Algebra

View full text Add to dashboard Cite

We introduce a new family of twisted generalized Weyl algebras, called multiparameter twisted Weyl algebras, for which we parametrize all simple quotients of a certain kind. Both Jordan's simple localization of the multiparameter quantized Weyl algebra and Hayashi's q-analog of the Weyl algebra are special cases of this construction. We classify all simple weight modules over any multiparameter twisted Weyl algebra. Extending results by Benkart and Ondrus, we also describe all Whittaker pairs up to isomorphism over a class of twisted generalized Weyl algebras which includes the multiparameter twisted Weyl algebras.

show abstract

Clifford and Weyl Superalgebras and Spinor Representations

Hartwig

Serganova

2019

Transformation Groups

View full text Add to dashboard Cite

We construct a family of twisted generalized Weyl algebras which includes Weyl-Clifford superalgebras and quotients of the enveloping algebras of gl(m|n) and osp(m|2n). We give a condition for when a canonical representation by differential operators is faithful. Lastly, we give a description of the graded support of these algebras in terms of pattern-avoiding vector compositions.2010 Mathematics Subject Classification. 17B35; 15A66.

show abstract

Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras

Cited by 10 publications

References 26 publications

Twisted Generalized Weyl Algebras and Primitive Quotients of Enveloping Algebras

Twisted Generalized Weyl Algebras and Primitive Quotients of Enveloping Algebras

Multiparameter twisted Weyl algebras

Clifford and Weyl Superalgebras and Spinor Representations

Contact Info

Product

Resources

About