A new class of associative algebras is introduced and studied. These algebras are related to simple complex Lie algebras (or root systems). Roughly speaking, they are finite dimensional approximations to the enveloping algebra U(g) viewed as a module over its center. It seems that several important questions on semisimple algebras and their representations can be formulated, studied and sometim...

We construct a generator system of the annihilator of a generalized Verma module of a classical reductive Lie algebra induced from a character of a parabolic subalgebra as an analogue of the minimal polynomial of a matrix. In a classical limit it gives a generator system of the defining ideal of any semisimple co-adjoint orbit of the Lie algebra. We also give some applications to integral geome...

A description of transitive actions of a semisimple algebraic group G on toric varieties is obtained. Every toric variety admitting such an action lies between a product of punctured affine spaces and a product of projective spaces. The result is based on the Cox realization of a toric variety as a quotient space of an open subset of a vector space V by a quasitorus action and on investigation ...

We show that actions of finite-dimensional semisimple commutative Hopf algebras H on //-module algebras A are essentially group-gradings. Moreover we show that the centralizer of H in the smash product A # H equals A" ® H. Using these we invoke results about group graded algebras and results about centralizers of separable subalgebras to give connections between the ideal structure of A, A and ...

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....