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1.
We classify infinitely generated projective modules over generalized Weyl algebras. For instance, we prove that over such algebras every projective module is a direct sum of finitely generated modules. 相似文献
2.
S. C. Coutinho 《Transactions of the American Mathematical Society》1997,349(8):3343-3352
In this paper we calculate some groups of singular modules over the complex Weyl algebra . In particular we determine conditions under which is an infinite dimensional vector space when or .
3.
For an algebraically closed field , we investigate a class of noncommutative -algebras called connected quantized Weyl algebras. Such an algebra has a PBW basis for a set of generators such that each pair satisfies a relation of the form , where and , with, in some sense, sufficiently many pairs for which . For such an algebra it turns out that there is a single parameter q such that each . Assuming that , we classify connected quantized Weyl algebras, showing that there are two types linear and cyclic. When q is not a root of unity we determine the prime spectra for each type. The linear case is the easier, although the result depends on the parity of n, and all prime ideals are completely prime. In the cyclic case, which can only occur if n is odd, there are prime ideals for which the factors have arbitrarily large Goldie rank.We apply connected quantized Weyl algebras to obtain presentations of two classes of quantum cluster algebras. Let be an odd integer. We present the quantum cluster algebra of a Dynkin quiver of type as a factor of a linear connected quantized Weyl algebra by an ideal generated by a central element. We also consider the quiver identified by Fordy and Marsh in their analysis of periodic quiver mutation. When n is odd, we show that the quantum cluster algebra of this quiver is generated by a cyclic connected quantized Weyl algebra in n variables and one further generator. We also present it as the factor of an iterated skew polynomial algebra in variables by an ideal generated by a central element. For both classes, the quantum cluster algebras are simple noetherian.We establish Poisson analogues of the results on prime ideals and quantum cluster algebras. We determine the Poisson prime spectra for the semiclassical limits of the linear and cyclic connected quantized Weyl algebras and show that, when n is odd, the cluster algebras of and are simple Poisson algebras that can each be presented as a Poisson factor of a polynomial algebra, with an appropriate Poisson bracket, by a principal ideal generated by a Poisson central element. 相似文献
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5.
The first purpose of this paper is to set up a general notion of skew power series rings S over a coefficient ring R, which are then studied by filtered ring techniques. The second goal is the investigation of the class of S-modules which are finitely generated as R-modules. In the case that S and R are Auslander regular we show in particular that the codimension of M as S-module is one higher than the codimension of M as R-module. Furthermore its class in the Grothendieck group of S-modules of codimension at most one less vanishes, which is in the spirit of the Gersten conjecture for commutative regular local rings. Finally we apply these results to Iwasawa algebras of p-adic Lie groups. 相似文献
6.
Stefan Gille 《manuscripta mathematica》2006,121(4):437-450
Let
be an Azumaya algebra over a locally noetherian scheme X. We describe in this work quasi-coherent
-bimodules which are injective in the category of sheaves of left
-modules 相似文献
7.
Yves Lequain 《Journal of Pure and Applied Algebra》2011,215(4):531-545
Let K be a field of characteristic zero, n≥1 an integer and An+1=K[X,Y1,…,Yn]〈∂X,∂Y1,…,∂Yn〉 the (n+1)th Weyl algebra over K. Let S∈An+1 be an order-1 differential operator of the type with ai,bi∈K[X] and gi∈K[X,Yi] for every i=1,…,n. We construct an algorithm that allows one to recognize whether S generates a maximal left ideal of An+1, hence also whether An+1/An+1S is an irreducible non-holonomic An+1-module. The algorithm, which is a powerful instrument for producing concrete examples of cyclic maximal left ideals of An, is easy to implement and quite useful; we use it to solve several open questions.The algorithm also allows one to recognize whether certain families of algebraic differential equations have a solution in K[X,Y1,…,Yn] and, when they have one, to compute it. 相似文献
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9.
In this paper, we give a complete classification of irreducible Harish-Chandra modules for any generalized Heisenberg-Virasoro algebra. In particular, we present a simpler and more conceptual proof of the classification of irreducible Harish-Chandra modules over the classical Heisenberg-Virasoro algebra, which was first obtained by Rencai Lu and Kaiming Zhao in [LZ1]. Our methods are based on the ideas of polynomial modules from [B1, BB]. 相似文献
10.
Vladimir Mazorchuk 《Compositio Mathematica》1999,115(1):21-35
We constuct and investigate a structure of Verma-like modules over generalized Witt algebras. We also prove Futorny-like theorem for irreducible weight modlues whose dimensions of the weight spaces are uniformly bounded. 相似文献
11.
Let Hc be the rational Cherednik algebra of type An-1 with spherical subalgebra Uc=eHce. Then Uc is filtered by order of differential operators, with associated graded ring where W is the nth symmetric group. We construct a filtered Z-algebra B such that, under mild conditions on c:• the category B-qgr of graded noetherian B-modules modulo torsion is equivalent to Uc-mod;• the associated graded Z-algebra has grB-lqgr?coh Hilb(n), the category of coherent sheaves on the Hilbert scheme of points in the plane.This can be regarded as saying that Uc simultaneously gives a non-commutative deformation of h⊕h*/W and of its resolution of singularities Hilb(n)→h⊕h*/W. As we show elsewhere, this result is a powerful tool for studying the representation theory of Hc and its relationship to Hilb(n). 相似文献
12.
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 . 相似文献
13.
Xiangui Zhao 《中国科学 数学(英文版)》2023,66(5):887-906
We study the growth and the Gelfand-Kirillov dimension(GK-dimension) of the 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, i.e., GKdim(A) is either 3 or ∞ in this case. Our results generalize several existing results in the literature and ca... 相似文献
14.
Let
be the Hecke algebra of the symmetric group
over a field K of characteristic
and
a primitive
-th root of one in K. We show that an
-module is projective if and only if its restrictions to any
-parabolic subalgebra of
is projective.
Moreover, we give a new construction of blocks of
-parabolic subalgebras, in terms of skew group algebras over local commutative
algebras.
Received: 30 June 2003 相似文献
15.
We define and investigate a class of Koszul quasi-hereditary algebras for which there is a natural equivalence between the bounded derived category of graded modules and the bounded derived category of graded modules over (a proper version of) the extension algebra of standard modules. Examples of such algebras include, in particular, the multiplicity free blocks of the BGG category O, and some quasi-hereditary algebras with Cartan decomposition in the sense of König. 相似文献
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V. S. Mazorchuk 《Ukrainian Mathematical Journal》1998,50(9):1461-1463
We classify unitarizable modules with highest weight and unitarizable modules of an intermediate series over generalized Virasoro
algebras.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1278–1280, September, 1998. 相似文献
18.
By means of a certain module V and its tensor powers in a finite tensor category, we study a question of whether the depth of a Hopf subalgebra R of a finite-dimensional Hopf algebra H is finite. The module V is the counit representation induced from R to H, which is then a generalized permutation module, as well as a module coalgebra. We show that if in the subalgebra pair either Hopf algebra has finite representation type, or V is either semisimple with R∗ pointed, projective, or its tensor powers satisfy a Burnside ring formula over a finite set of Hopf subalgebras including R, then the depth of R in H is finite. One assigns a nonnegative integer depth to V, or any other H-module, by comparing the truncated tensor algebras of V in a finite tensor category and so obtains upper and lower bounds for depth of a Hopf subalgebra. For example, a relative Hopf restricted module has depth 1, and a permutation module of a corefree subgroup has depth less than the number of values of its character. 相似文献
19.
Stefan Veldsman 《代数通讯》2013,41(8):3659-3673
Since their introduction in 1964 as a combinatorial tool, incidence algebras have been studied in their own right. In particular, the Jacobson and nilradicals of incidence algebras over commutative rings with identity were determined.Here we present the general radical theory for incidence algebras, with the emphasis on hypernilpotent and subidempotent radicals. 相似文献
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