On cell modules of symmetric cellular algebras

Let A be a symmetric cellular algebra with cell datum (??, M, C, i) and let ${\Lambda_1=\{\lambda \in \Lambda_0 \mid W(\lambda) \, {\rm is \, simple}\}}$ . We prove that ??₁ consists of two parts: one gives a lower bound for the cardinality of the set of cell modules with zero bilinear forms and the other parametrizes all the projective cell modules. Moreover, it is proved in Li (arxiv: math0911.3524, 2009) that the dual basis of ${\{C_{S, T}^{\lambda} \mid \lambda \in \Lambda, S,T \in M(\lambda)\}}$ is again cellular. In this paper, we will study the cell modules defined by dual basis. In particular, we study the dual basis of the Murphy basis.     

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.     

On simple modules over conformal Galilei algebras

We study irreducible representations of two classes of conformal Galilei algebras in 1-spatial dimension. We construct a functor which transforms simple modules with nonzero central charge over the Heisenberg subalgebra into simple modules over the conformal Galilei algebras. This can be viewed as an analogue of oscillator representations. We use oscillator representations to describe the structure of simple highest weight modules over conformal Galilei algebras. We classify simple weight modules with finite dimensional weight spaces over finite dimensional Heisenberg algebras and use this classification and properties of oscillator representations to classify simple weight modules with finite dimensional weight spaces over conformal Galilei algebras.     

On the structure of periodic modules over tame algebras

We describe the structure of stable tubes in the Auslander-Reiten quivers of tame algebras formed by indecomposable modules which do not lie on infinite short cycles. In particular, we prove that all algebras whose module categories have no infinite short cycles are of linear growth.

     

On injective modules over Azumaya algebras over locally noetherian schemes

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     

Rigid modules over preprojective algebras

Let Λ be a preprojective algebra of simply laced Dynkin type Δ. We study maximal rigid Λ-modules, their endomorphism algebras and a mutation operation on these modules. This leads to a representation-theoretic construction of the cluster algebra structure on the ring ℂ[N] of polynomial functions on a maximal unipotent subgroup N of a complex Lie group of type Δ. As an application we obtain that all cluster monomials of ℂ[N] belong to the dual semicanonical basis. Mathematics Subject Classification (2000) 14M99, 16D70, 16E20, 16G20, 16G70, 17B37, 20G42     

On tense modules for extended algebras over rational fields

Let k(x) be the field of fractions of the polynomial algebra k[x] over the field k. We prove that, for an arbitrary finite dimensional k-algebra Λ, any finitely generated Λ ⊗_k k(x)-module M such that its minimal projective presentation admits no non-trivial selfextension is of the form M ≅ N^k(x), for some finitely generated Λ-module N. Some consequences are derived for tilting modules over the rational algebra Λ ⊗_k k(x) and for some generic modules for Λ. Received: 24 November 2003; revised: 11 February 2005     

Twistors for modules over algebras

We study the concept of module twistor for a module over an algebra. This concept provides a unifying framework for various deformed constructions of modules over algebras, such as module R-matrices, (n-factor iterated) twisted tensor products and L-R-twisted tensor products of algebras. Among the main results, we find the relations among these constructions. Furthermore, we study some properties of module twistors.     

Quasi-Whittaker modules over the conformal Galilei algebras

In this paper, we study quasi-Whittaker modules over the conformal Galilei algebras. We determine all quasi-Whittaker vectors and the simplicities of the universal quasi-Whittaker modules. For the reducible ones, we determine the corresponding simple quotient modules. Thus we classify all simple quasi-Whittaker modules for the conformal Galilei algebras.     

On the vanishing of extensions of modules over reduced enveloping algebras

Supported by NSF grants DMS-9001689 and DMS-9301929     

On derived categories of modules over 2-vertex basic algebras

We investigate finite dimensional 2-vertex basic algebras of finite global dimension and the derived categories of modules over such algebras. We prove that any superrigid object in the derived category of modules over a “loop-kind” two-vertex algebra is a pure module up to the action of Serre functor and translation. All superrigid objects in the derived categories of modules over two-vertex algebras of global dimension 2 are described. Also we obtain a complete classification of two-vertex basic algebras possessing a full exceptional pair in the derived category of modules.     

On supports and associated primes of modules over the enveloping algebras of nilpotent Lie algebras

Let be a nilpotent Lie algebra, over a field of characteristic zero, and its universal enveloping algebra. In this paper we study: (1) the prime ideal structure of related to finitely generated -modules , and in particular the set of associated primes for such (note that now is equal to the set of annihilator primes for ); (2) the problem of nontriviality for the modules when is a (maximal) prime of , and in particular when is the augmentation ideal of . We define the support of , as a natural generalization of the same notion from commutative theory, and show that it is the object of primary interest when dealing with (2). We also introduce and study the reduced localization and the reduced support, which enables to better understand the set . We prove the following generalization of a stability result given by W. Casselman and M. S. Osborne in the case when , as in the theorem, are abelian. We also present some of its interesting consequences.

Theorem. Let be a finite-dimensional Lie algebra over a field of characteristic zero, and an ideal of ; denote by the universal enveloping algebra of . Let be a -module which is finitely generated as an -module. Then every annihilator prime of , when is regarded as a -module, is -stable for the adjoint action of on .

     

On minimal disjoint degenerations of modules over tame path algebras

For representations of tame quivers the degenerations are controlled by the dimensions of various homomorphism spaces. Furthermore, there is no proper degeneration to an indecomposable. Therefore, up to common direct summands, any minimal degeneration from M to N is induced by a short exact sequence 0→U→M→V→0 with indecomposable ends that add up to N. We study these ‘building blocs’ of degenerations and we prove that the codimensions are bounded by two. Therefore, a quiver is Dynkin resp. Euclidean resp. wild iff the codimension of the building blocs is one resp. bounded by two resp. unbounded. We explain also that for tame quivers the complete classification of all the building blocs is a finite problem that can be solved with the help of a computer.     

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].     

Doi-hopf modules over weak hopf algebras

The theorv of Doi-Hopf modules [8,11] is generalized to Weak Hopf Algebras [1, 14, 2].     

Projective cell modules of Frobenius cellular algebras

In this note we give a criterion of projectiveness of the simple cell modules over finite dimensional Frobenius cellular algebras.     

Extensions of modules over Weyl algebras

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 .