d-critical modules of length 2 over weyl algebras

We construct a family of modules over Weyl algebras with the property of being non-simple of finite length and alsod-critical (i.e.d(M)>d(M/N) for every non-trivial submoduleN, ofM). Hered stands for the Gelfand-Kirillov dimension. We further study some properties of these modules.     

Dimensions of irreducible modules over twisted group algebras

LetN be a normal subgroup of a finite groupG, letF be an algebraically closed field, letZ ²(G, F ^*) and letV be an irreducible module over the twisted group algebraF . If charF=p>0 divides (GN), assume thatG/N isp-solvable. It is proved that dim _F V divides (GN)d whered is the dimension of an irreducible constituent ofV _N. The special case where=1 andN is abelian yields a well-known theorem of Dade [3]. Another special case, namely whereN is abelian, charF(GN) and the restriction of ofNxN is a coboundary is a generalization of the main result of Ng [5].     

Verma modules over generalized Witt algebras

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.     

Harish-Chandra modules over generalized Heisenberg-Virasoro algebras

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].     

On unitarizable modules over generalized Virasoro algebras

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.     

Simple modules over generalized matrix rings

We study simple modules over the ring of generalized matrices. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 245–247, 2007.     

Indecomposable representations of generalized weyl algebras

For a class of generalized Weyl algebras which includes the Weyl algebras A_n a criterion is given as to when the category of indecomposable weight and generalized weight modules with supports from a fixed orbit is tame. In tame cases indecomposable modules are described.     

Pseudo-unitarizable weight modules over generalized Weyl algebras

We define a notion of pseudo-unitarizability for weight modules over a generalized Weyl algebra (of rank one, with commutative coefficient ring R), which is assumed to carry an involution of the form X^∗=Y, R^∗⊆R. We prove that a weight module V is pseudo-unitarizable iff it is isomorphic to its finitistic dual V^?. Using the classification of weight modules by Drozd, Guzner and Ovsienko, we obtain necessary and sufficient conditions for an indecomposable weight module to be isomorphic to its finitistic dual, and thus to be pseudo-unitarizable. Some examples are given, including U_q(sl₂) for q a root of unity.     

Non-finitely generated projective modules over generalized Weyl algebras

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.     

Simple lie color algebras of weyl type

A class of graded simple associative algebras are constructed, and from them, simple Lie color algebras are obtained. The structure of these simple Lie color algebras is explicitly described. More precisely, for an (ε, Γ)-color-commutative associative algebraA with an identity element over a fieldF of characteristic not 2, and for a color-commutative subalgebraD of color-derivations ofA, denote byA[D] the associative subalgebra of End (A) generated byA (regarded as operators onA via left multiplication) andD. It is easily proved that, as an associative algebra,A[D] is Γ-graded simple if and only ifA is Γ-gradedD-simple. SupposeA is Γ-gradedD-simple. Then, (a)A[D] is a free leftA-module; (b) as a Lie color algebra, the subquotient [A[D],A[D]]/Z(A[D])∩[A[D],A[D]] is simple (except one minor case), whereZ(A[D]) is the color center ofA[D]. This work was supported by NSF of China, National Educational Department of China, Jiangsu Educational Committee, and Hundred Talents Program of Chinese Academy of Sciences. These authors were partially supported by Academy of Mathematics and System Sciences during their visit to this academy.     

Left almost split morphisms and generalized weyl algebras

It is proved that the category of modules of finite length over a broad class of generalized Weyl algebras contains no left almost split morphism starting from a simple module. It is shown that a similar assertion holds for the algebraUsl₂(k) over an algebraically closed field k of characteristic 0. As a by-product, a new series of simple modules for such algebras is constructed. Translated fromMatematicheskie Zametki, Vol. 66, No. 5 pp. 734–740, November, 1999.     

Generalized Weyl modules for twisted current algebras

We introduce the notion of generalized Weyl modules for twisted current algebras. We study their representation-theoretic and combinatorial properties and also their connection with nonsymmetric Macdonald polynomials. As an application, we compute the dimension of the classical Weyl modules in the remaining unknown case.     

Weyl modules for the twisted loop algebras

The notion of a Weyl module, previously defined for the untwisted affine algebras, is extended here to the twisted affine algebras. We describe an identification of the Weyl modules for the twisted affine algebras with suitably chosen Weyl modules for the untwisted affine algebras. This identification allows us to use known results in the untwisted case to compute the dimensions and characters of the Weyl modules for the twisted algebras.     

拟遗传代数的诱导模与广义Betti数

本文引入了模的广义Betti数,给出了经典Betti数与广义Betti数之间的关系,证明了具有纯粹强正合Borel子代数的零关系拟遗传代数的诱导模的极小投射分解可通过其正合Borel子代数的相应模的极小投射分解的诱导给出,从而两者具有相同的广义Betti数.     

Smash product algebras over twisted dimodule algebras

Results of the research for smash product algebras over dimodule algebras are generalized to the more general twisted dimodule algebras.     

Verma modules over generalized Virasoro algebras Vir[G]

Let F be a field of characteristic 0, not necessarily algebraically closed, and G be an additive subgroup of F. For any total order on G which is compatible with the group addition, and for any , a Verma module over the generalized Virasoro algebra Vir[G] is defined. In the present paper, the irreducibility of Verma modules is completely determined.