Modules for double affine Lie algebras

Imaginary Verma modules, parabolic imaginary Verma modules,and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investigated,and several results are generalized from the affine Lie algebras. In particular,imaginary highest weight modules, integrable modules, and irreducibility criterion are also studied.     

Gelfand-Kirillov Dimension and Reducibility of Scalar Generalized Verma Modules

The Gelfand-Kirillov dimension is an invariant which can measure the size of infinite-dimensional algebraic structures. In this article, we show that it can also measure the reducibility of scalar generalized Verma modules. In particular, we use it to determine the reducibility of scalar generalized Verma modules associated with maximal parabolic subalgebras in the Hermitian symmetric case.     

Quantum Deformations of α-Stratified Modules

We construct quantum analogues of a class of generalized Verma modulesinduced from nonsolvable parabolic subalgebras of simple Lie algebras. Weshow that these quantum modules are true deformations of the underlyingclassical modules in the sense that the weight-space decomposition ispreserved.     

ω-Gorenstein Modules

We introduce the notion of ω-Gorenstein modules, where ω is a faithfully balanced self-orthogonal module. This gives a common generalization of both Gorenstein projective modules and Gorenstein injective modules. We consider such modules in the tilting theory. Consequently, some results due to Auslander and colleagues and Enochs and colleagues are generalized.     

Recursive properties of branching and BGG resolution

Recurrence relations for branching coefficients are based on a certain decomposition of the singular element. We show that this decomposition can be used to construct parabolic Verma modules and to obtain the generalized Weyl-Verma formulas for characters. We also demonstrate that the branching coefficients determine the generalized Bernstein-Gelfand-Gelfand resolution.     

Doi–Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras

For a quasi-Hopf algebra H, a left H-comodule algebra  and a right H-module coalgebra C we will characterize the category of Doi–Hopf modules ^C ?(H) in terms of modules. We will also show that for an H-bicomodule algebra  and an H-bimodule coalgebra C the category of generalized Yetter–Drinfeld modules (H) ^C is isomorphic to a certain category of Doi–Hopf modules. Using this isomorphism we will transport the properties from the category of Doi–Hopf modules to the category of generalized Yetter–Drinfeld modules.     

可积模的权

本文定义了Ｋａｃ－Ｍｏｏｄｙ代数的一个新的可积模范畴，并且给出了一个可积模是否属于这个模范畴的一个判别准则．另外还详细研究了这个模范畴中的可积模的权系。特别我们定义了虚权和实权。还详细地计算了一些模的虚权和实权，还给出了双曲型广义Ｃａｒｔａｎ矩阵的新刻划．这使我们能够计算一些双曲型Ｋａｃ－Ｍｏｏｄｙ代数的可积模的权。     

Gorenstein n-X-Injective and n-X-Flat Modules with Respect to a Special Finitely Presented Module

Let R be a ring,X a class of R-modules and n ≥ 1 an integer.We intro-duce the concepts of Gorenstein n-X-injective and n-X-flat modules via special finitely presented modules.Besides,we obtain some equivalent properties of these modules on n-X-coherent rings.Then we investigate the relations among Gorenstein n-X-injective,n-X-flat,injective and fiat modules on X-FC-rings (i.e.,self n-X-injective and n-X-coherent rings).Several known results are generalized to this new context.     

On generalized k-syzygy modules

In this paper, we first introduce the notion of generalized k-syzygy modules, and then give an equivalent characterization that the class of generalized k-syzygy modules coincides with that ofω-k-torsionfree modules. We further study the extension closure of the category consisting of generalized k-syzygy modules. Some known results are obtained as corollaries.     

Verma type modules of level zero for affine Lie algebras

We study the structure of Verma type modules of level zero induced from non-standard Borel subalgebras of an affine Kac-Moody algebra. For such modules in ``general position' we describe the unique irreducible quotients, construct a BGG type resolution and prove the BGG duality in certain categories. All results are extended to generalized Verma type modules of zero level.

     

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.     

Cuspidal systems for affine Khovanov–Lauda–Rouquier algebras

A cuspidal system for an affine Khovanov–Lauda–Rouquier algebra $R_\alpha $ yields a theory of standard modules. This allows us to classify the irreducible modules over $R_\alpha $ up to the so-called imaginary modules. We describe minuscule imaginary modules, laying the groundwork for future study of imaginary Schur–Weyl duality. We introduce colored imaginary tensor spaces and reduce a classification of imaginary modules to one color. We study the characters of cuspidal modules. We show that under the Khovanov–Lauda–Rouquier categorification, cuspidal modules correspond to dual root vectors.     

On the Reducibility of Scalar Generalized Verma Modules of Abelian Type

A parabolic subalgebra \(\mathfrak {p}\) of a complex semisimple Lie algebra \(\mathfrak {g}\) is called a parabolic subalgebra of abelian type if its nilpotent radical is abelian. In this paper, we provide a complete characterization of the parameters for scalar generalized Verma modules attached to parabolic subalgebras of abelian type such that the modules are reducible. The proofs use Jantzen’s simplicity criterion, as well as the Enright-Howe-Wallach classification of unitary highest weight modules.     

几乎有限表现模

利用几乎有限表现模来刻划凝聚环和半遗传环.通过讨论几乎有限表现模和广义有限表现模之间的关系,得出了几个关于几乎有限表现模和凝聚环、半遗传环的等价条件,改进了已有的结论,把刻划凝聚环的模缩小到几乎有限表现模.     

Monomial Modules and Endo-Monomial Modules

We determine the multiplicity algebras and multiplicity modules of a p-monomial module. For a general p-group P, we find a sufficient and necessary condition for an endo-monomial P-module to be an endo-permutation P-module, and prove that a capped indecomposable endo-monomial P-module is of p ^′-rank. At last, we give an alternative definition of the generalized Dade P-group.     

Central Extensions of Precrossed Modules

We classify the precrossed module central extensions using the second cohomology group of precrossed modules. We relate these central extensions to the relative central group extensions of Loday, and to other notions of centrality defined in general contexts. Finally we establish a Universal Coefficient Theorem for the (co)homology of precrossed modules, which we use to describe the precrossed module central extensions in terms of the generalized Galois theory developed by Janelidze.     

Graded Modules Over the q-Analog Virasoro-Like Algebra

In this paper, we deal with the classification of the irreducible Z-graded and Z ²-graded modules with finite dimensional homogeneous subspaces for the q analog Virasoro-like algebra L. We first prove that a Z-graded L-module must be a uniformly bounded module or a generalized highest weight module. Then we show that an irreducible generalized highest weight Z-graded module with finite dimensional homogeneous subspaces must be a highest (or lowest) weight module and give a necessary and sufficient condition for such a module with finite dimensional homogeneous subspaces. We use the Z-graded modules to construct a class of Z ²-graded irreducible generalized highest weight modules with finite dimensional homogeneous subspaces. Finally, we classify the Z ²-graded L-modules. We first prove that a Z ²-graded module must be either a uniformly bounded module or a generalized highest weight module. Then we prove that an irreducible nontrivial Z ²-graded module with finite dimensional homogeneous subspaces must be isomorphic to a module constructed as above. As a consequence, we also classify the irreducible Z-graded modules and the irreducible Z ²-graded modules with finite dimensional homogeneous subspaces and center acting nontrivial. Supported by the National Science Foundation of China (No 10671160), the China Postdoctoral Science Foundation (No. 20060390693), the Specialized Research fund for the Doctoral Program of Higher Education (No.20060384002), and the New Century Talents Supported Program from the Education Department of Fujian Province.     

Some Remarks on Generalized GCD Domains

An integral domain R is a generalized GCD (GGCD) domain if the semigroup of invertible ideals of R is closed under intersection. In this article we extend the definition of PF-prime ideals to GGCD domains and develop a theory of these ideals which allows us to characterize Prüfer and π -domains among GGCD domains. We also introduce the concept of generalized GCD modules as a natural generalization of GGCD domains to the module case. An R-module M is a GGCD module if the set of invertible submodules of M is closed under intersection. We show that an integral domain R is a GGCD domain if and only if a faithful multiplication R-module M is a GGCD module. Various properties and characterizations of faithful multiplication GGCD modules over integral domains are considered and consequently, necessary and sufficient conditions for a ring R(M), the idealization of M, to be a GGCD ring are given.     

Verma type modules for toroidal lie algebras

We study the structure of imaginary Verma modules induced from the"natural"Borel subalgebra of a toroidal Lie algebra. In particular, we establish a criterion of irreducibility for imaginary Verma modules and describe their submodules and irreducible quotients. We also describe the structure of Verma type modules in the case of sl(2)-toroidal Lie algebra over two variables.