Introduction. The purpose of this paper is to compare homological properties of an analytic representation of a semisimple Lie group and of its Harish-Chandra module. Throughout the paper G0 denotes a connected semisimple Lie group with finite center. Let π be an admissible representation of G0 on a complete, locally convex Hausdorff topological vector space Mπ . Vectors in Mπ transforming fini...

Let Λ be a preprojective algebra of Dynkin type, and let G be the corresponding complex semisimple simply connected algebraic group. We study rigid modules in subcategories Sub Q for Q an injective Λ-module, and we introduce a mutation operation between complete rigid modules in Sub Q. This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to G.

Let g be a complex semisimple Lie algebra, let h ⊂ b be respectively Cartan and Borel subalgebras of g, put n := [b, b], say that the roots of h in n are positive, let W be the Weyl group equipped with the Bruhat ordering, let O 0 be the category of those BGG-modules which have the generalized infinitesimal character of the trivial module. The simple objects of O 0 are parametrized by W.

Introduction This is a continuation of the investigation in [L3] into the module structure of those modules for quantum enveloping algebras U analogous to the coho-mology groups of line bundles on the flag varieties for semisimple algebraic groups. In [L3], we used the admissible categories for Hopf algebras and their induction theory and proved a modified Borel-Bott-Weil theorem for R l(w) Ind

A condition on an affine central subalgebra Z of a noetherian algebra A of finite GelfandKirillov dimension, which we call here unruffledness, is shown to be equivalent in some circumstances to the flatness of A as a Z-module. Unruffledness was studied by Borho and Joseph in work on enveloping algebras of complex semisimple Lie algebras, and we discuss applications of our result to enveloping a...

We study the tensor category of modules over a semisimple bialgebra H under the assumption that irreducible H -modules of the same dimension >1 are isomorphic. We consider properties of Clebsch–Gordan coefficients showing multiplicities of occurrences of each irreducible H -module in a tensor product of irreducible ones. It is shown that, in general, these coefficients cannot have small values....

Let T be a Noetherian ring and f a nonzerodivisor on T . We study concrete necessary and sufficient conditions for a module over R = T/(f) to be weakly liftable to T , in the sense of Auslander, Ding and Solberg. We focus on cyclic modules and get various positive and negative results on the lifting and weak lifting problems. For a module over T we define the loci for certain properties: liftab...

The notions of pure ideals and pure submodules play an important role in the study of rings and their module categories. In this paper, we introduce right t-pure fuzzy ideals and pure fuzzy submodules of an R-module. We establish their basic properties, and use them to obtain fuzzy theoretic characterizations of right weakly regular rings. Among other results, we prove that a ring R is right we...

A new class of locally unital and finite dimensional algebras over an arbitrary algebraically closed field is discovered. Each them admits upper weakly triangular decomposition, a generalization split decomposition. It established that the category -lfdmod left -modules fully stratified in sense Brundan-Stroppel. Moreover, semisimple if only its centralizer subalgebras associated to certain ide...