The quantum euclidean algebra and its prime spectrum

Bavula, V. V.; Lü, Tao

doi:10.1007/s11856-017-1503-1

Cited by 5 publications

(3 citation statements)

References 11 publications

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“…A quantum analog of U , the quantum Euclidean algebra, was defined and studied in Ref. 6 where its prime, completely prime, primitive, and maximal ideals were classified. Classification of prime ideals of U .…”

Section: Introductionmentioning

confidence: 99%

Prime ideals of the enveloping algebra of the Euclidean algebra and a classification of its simple weight modules

Bavula

Lü

2017

Journal of Mathematical Physics

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A classification of the simple weight modules is given for the (6-dimensional) Euclidean Lie algebra e(3) = sl 2 ⋉ V 3 . As an intermediate step, a classification of all simple modules is given for the centralizer C of the Cartan element H (in the universal enveloping algebra U = U(e(3))). Generators and defining relations for the algebra C are found (there are three quadratic relations and one cubic relation). The algebra C is a Noetherian domain of Gelfand-Kirillov dimension 5. Classifications of prime, primitive, completely prime, and maximal ideals are given for the algebra U. Published by AIP Publishing. [http://dx

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Section: Introductionmentioning

confidence: 99%

Prime ideals of the enveloping algebra of the Euclidean algebra and a classification of its simple weight modules

Bavula

Lü

2017

Journal of Mathematical Physics

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“…Many algebras of common interest fall into this class, for instance, Weyl algebras, certain quantum enveloping algebras, some iterated Ore extensions, noetherian (generalized) down-up algebras, Heisenberg algebras, etc. (see [7,12,55] for more examples). GWAs were introduced by Bavula [7] in early 1990s.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Growth of generalized Weyl algebras over polynomial algebras and Laurent polynomial algebras

Zhao

2022

Preprint

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We mainly study the growth and Gelfand-Kirillov dimension (GK-dimension) of generalized Weyl algebra (GWA) A = D(σ, a) where D is a polynomial algebra or a Laurent polynomial algebra. Several necessary and sufficient conditions for GKdim(A) = GKdim(D) + 1 are given. In particular, we prove a dichotomy of the GK-dimension of GWAs over the polynomial algebra in two indeterminates, namely, GKdim(A) is either 3 or ∞ in this case. Our results generalize several ones in the literature and can be applied to determine the growth, GK-dimension, simplicity, and cancellation properties of some GWAs.

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“…Classifications of (various classes of) simple weight modules over algebras that are close to the (generalized) Weyl algebras are given in [28,18,26,32,27,22,29,23,11].…”

Section: Introductionmentioning

confidence: 99%

Indecomposable generalized weight modules over the algebra of polynomial integro-differential operators

Bavula

Bekkert

Futorny

2018

Proc. Amer. Math. Soc.

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For the algebra In = K x 1 , . . . , xn, ∂ 1 , . . . , ∂n, 1 , . . . , n of polynomial integrodifferential operators over a field K of characteristic zero, a classification of simple weight and generalized weight (left and right) In-modules is given. It is proven that the category of weight In-modules is semisimple. An explicit description of generalized weight In-modules is given and using it a criterion is obtained for the problem of classification of indecomposable generalized weight In-modules to be of finite representation type, tame or wild. In the tame case, a classification of indecomposable generalized weight In-modules is given. In the wild case 'natural' tame subcategories are considered with explicit description of indecomposable modules. It is proven that every generalized weight In-module is a unique sum of absolutely prime modules. For an arbitrary ring R, we introduce the concept of absolutely prime R-module (a nonzero R-module M is absolutely prime if all nonzero subfactors of M have the same annihilator). It is shown that every indecomposable generalized weight In-module is equidimensional. A criterion is given for a generalized weight In-module to be finitely generated.

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The quantum euclidean algebra and its prime spectrum

Abstract: For the quantum Euclidean algebra, its prime, completely prime and maximal spectra are described (together with inclusions of prime ideals). The centre is generated by two algebraically independent elements (one is quadratic and the other is cubic).

Cited by 5 publications

References 11 publications

Prime ideals of the enveloping algebra of the Euclidean algebra and a classification of its simple weight modules

Prime ideals of the enveloping algebra of the Euclidean algebra and a classification of its simple weight modules

Growth of generalized Weyl algebras over polynomial algebras and Laurent polynomial algebras

Indecomposable generalized weight modules over the algebra of polynomial integro-differential operators

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