Isomorphism problems and groups of automorphisms for generalized Weyl algebras

Bavula, V. V.; Jordan, David A.

doi:10.1090/s0002-9947-00-02678-7

Cited by 93 publications

(85 citation statements)

References 30 publications

(59 reference statements)

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“…In [4] and [16] the isomorphism problem was solved for quantum generalized Weyl algebras over a Laurent polynomial ring. We now solve the isomorphism problem for k[h ±1 ](σ inv , a) for a ∈ k[h ±1 ], completing the classification of isomorphisms for the noncommative families in Proposition 1.2.…”

Section: Isomorphismsmentioning

confidence: 99%

Endomorphisms of Quantum Generalized Weyl Algebras

Kitchin

Launois

2014

Lett Math Phys

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Abstract. We prove that every endomorphism of a simple quantum generalized Weyl algebra A over a commutative Laurent polynomial ring in one variable is an automorphism. This is achieved by obtaining an explicit classification of all endomorphisms of A. Our main result applies to minimal primitive factors of the quantized enveloping algebra U q (sl 2 ) and certain minimal primitive quotients of the positive part of U q (so 5 ).Mathematics Subject Classification (2010). 16W35, 16S32, 16W20, 17B37.

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Section: Isomorphismsmentioning

confidence: 99%

Endomorphisms of Quantum Generalized Weyl Algebras

Kitchin

Launois

2014

Lett Math Phys

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“…A e (A e ) 3 (A e ) 4 (A e ) 4 (A e ) We are so fortunate that all the basis elements in Proposition 4 belong to C (1) • (A, A ν ), and hence Theorem 6 works. Applying the Connes operator B to these elements, we have…”

Section: Define S As Followsmentioning

confidence: 91%

“…So far, a lot of algebraic properties of generalized Weyl algebras have been revealed, such as irreducible representations, homological dimensions, isomorphisms, and automorphisms (cf. [1,2,4,18]). In particular, the Hochschild homology and cohomology for generalized Weyl algebras A over a polynomial algebra in one variety have been computed [8,20].…”

Section: Introductionmentioning

confidence: 99%

Batalin-Vilkovisky algebra structures on Hochschild cohomology of generalized Weyl algebras

Liu

Wen

2021

Front. Math

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We devote to the calculation of Batalin-Vilkovisky algebra structures on the Hochschild cohomology of skew Calabi-Yau generalized Weyl algebras. We first establish a Van den Bergh duality at the level of complex. Then based on the results of Solotar et al., we apply Kowalzig and Krähmer's method to the Hochschild homology of generalized Weyl algebras, and translate the homological information into cohomological one by virtue of the Van den Bergh duality, obtaining the desired Batalin-Vilkovisky algebra structures. Finally, we apply our results to quantum weighted projective lines and Podleś quantum spheres, and the Batalin-Vilkovisky algebra structures for them are described completely.

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“…This result was obtained in [6] using different methods but here we wish to underline the common features and properties of the locally nilpotent derivations of the algebras A h as a whole, showing how they fit into the approach used by Dixmier and Rentschler in [10] and [18], respectively, and how their structure under the action of the group G h allows for the description of their automorphism groups. Another example of this phenomenon occurs in [2], where the authors study the isomorphisms and automorphisms of a family of generalized Weyl algebras over a polynomial algebra of rank one.…”

Section: The Locally Nilpotent Derivations Of a H In Characteristicmentioning

confidence: 99%

Actions of the additive group G_a on certain noncommutative deformations of the plane

Kaygorodov

Lopes

Mashurov

2021

Communications in Mathematics

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We connect the theorems of Rentschler [18] and Dixmier [10] on locally nilpotent derivations and automorphisms of the polynomial ring A 0 and of the Weyl algebra A 1, both over a field of characteristic zero, by establishing the same type of results for the family of algebras A h = 〈 x , y | y x − x y = h ( x ) 〉 , {A_h} = \left\langle {x,y|yx - xy = h\left( x \right)} \right\rangle , , where h is an arbitrary polynomial in x. In the second part of the paper we consider a field 𝔽 of prime characteristic and study 𝔽[t]-comodule algebra structures on Ah . We also compute the Makar-Limanov invariant of absolute constants of Ah over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of Ah .

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Isomorphism problems and groups of automorphisms for generalized Weyl algebras

Cited by 93 publications

References 30 publications

Endomorphisms of Quantum Generalized Weyl Algebras

Endomorphisms of Quantum Generalized Weyl Algebras

Batalin-Vilkovisky algebra structures on Hochschild cohomology of generalized Weyl algebras

Actions of the additive group G_a on certain noncommutative deformations of the plane

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Isomorphism problems and groups of automorphisms for generalized Weyl algebras

Cited by 93 publications

References 30 publications

Endomorphisms of Quantum Generalized Weyl Algebras

Endomorphisms of Quantum Generalized Weyl Algebras

Batalin-Vilkovisky algebra structures on Hochschild cohomology of generalized Weyl algebras

Actions of the additive group Ga on certain noncommutative deformations of the plane

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Product

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Actions of the additive group G_a on certain noncommutative deformations of the plane