The prime spectrum of the algebra $\mathbb K_q[X,Y]\rtimes U_q(\mathfrak {sl}_2)$ and a classification of simple weight modules

Bavula, V. V.; Lü, Tao

doi:10.4171/jncg/294

Cited by 5 publications

(2 citation statements)

References 15 publications

(25 reference statements)

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“…In fact, in [5] in the tame case a classification of indecomposable generalized weight modules was obtained for a large class of algebras, the, socalled, generalized Weyl algebras (the universal enveloping algebra U (sl 2 ) and the Weyl algebras are examples of generalized Weyl algebras as well as many other quantum groups are). Recently, for some non-semisimple Lie algebras G and their quantum analogues, classifications of simple weight modules are given: the Schrödinger algebra, [15]; sl 2 ⋉ V 2 , [16], where V 2 is the simple 2dimensional sl 2 -module; the enveloping algebra of the Euclidean algebra, [14]; K q [X, Y ] ⋊ U q (sl 2 ), [17]; the quantum spatial ageing algebra, [13].…”

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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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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“…The subalgebra generated by X , Y , Z is called the quantum Weyl algebra H q , see [2]. All simple weight modules over U q (s) with zero action of Z were classified in [4].…”

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confidence: 99%

BGG Category for the Quantum Schrödinger Algebra

Liu¹,

Li²

2020

Glasgow Math. J.

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In this paper, we study the BGG category O for the quantum Schrödinger algebra U q (s), where q is a nonzero complex number which is not a root of unity. If the central chargeż = 0, using the module Bż over the quantum Weyl algebra H q , we show that there is an equivalence between the full subcategory O[ż] consisting of modules with the central chargeż and the BGG category O (sl2) for the quantum group U q (sl 2 ). In the case thatż = 0, we study the subcategory A consisting of finite dimensional U q (s)-modules of type 1 with zero action of Z. Motivated by the ideas in [DLMZ, Mak], we directly construct an equivalent functor from A to the category of finite dimensional representations of an infinite quiver with some quadratic relations. As a corollary, we show that the category of finite dimensional U q (s)-modules is wild.

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On representations of the centrally extended Heisenberg double of SL2

Tao

2021

Journal of Mathematical Physics

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For the centrally extended Heisenberg double of SL2, its center is determined, the central factor algebras are described, and classifications of simple Harish-Chandra modules, simple Whittaker modules, and simple quasi-Whittaker modules are obtained. Two classes of simple weight modules with infinite-dimensional weight spaces are given. We also give a classification of simple modules that decompose into a direct sum of simple finite-dimensional sl2-modules with finite multiplicities.

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The prime spectrum of the algebra $\mathbb K_q[X,Y]\rtimes U_q(\mathfrak {sl}_2)$ and a classification of simple weight modules

Abstract: This is a repository copy of The prime spectrum of the algebra Kq[X,Y] Uq(sl2) and a ⋊ classification of simple weight modules.

Cited by 5 publications

References 15 publications

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

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

BGG Category for the Quantum Schrödinger Algebra

On representations of the centrally extended Heisenberg double of SL2

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