Let G be a simply connected semisimple algebraic group over an algebraically closed field k of characteristic 0 and let V be a rational simple G-module. If G/H ⊂ P(V ) is a spherical orbit, set X = G/H ⊂ P(V ) its closure. Then we describe the orbits of X and of its normalization e X in terms of spherical systems and we give necessary and sufficient conditions so that the normalization e X → X ...

We derive necessary and sufficient conditions for an ambiskew polynomial ring to have a Hopf algebra structure of a certain type. This construction generalizes many known Hopf algebras, for example U(sl2), Uq(sl2) and the enveloping algebra of the 3-dimensional Heisenberg Lie algebra. In a torsion-free case we describe the finite-dimensional simple modules, in particular their dimensions and pr...

Let G be a semisimple connected linear algebraic group over C, and X a wonderful G-variety. We study the possibility of realizing X as a closed subvariety of the projective space of a simple G-module. For any wonderful variety X we give a necessary and sufficient condition in terms of its set of spherical roots, which are certain characters of B (a fixed Borel subgroup of G) associated to the G...

In this article we describe the G×G-equivariant K-ring of X, where X is a regular compactification of a connected complex reductive algebraic group G. Furthermore, in the case when G is a semisimple group of adjoint type, and X its wonderful compactification, we describe its ordinary K-ring K(X). More precisely, we prove that K(X) is a free module over K(G/B) of rank the cardinality of the Weyl...

Let g be a semisimple complex Lie algebra and k ⊂ g be any algebraic subalgebra reductive in g. For any simple finite dimensional k-module V , we construct simple (g, k)-modules M with finite dimensional k-isotypic components such that V is a k-submodule of M and the Vogan norm of any simple k-submodule V ′ ⊂ M, V ′ 6≃ V , is greater than the Vogan norm of V . The (g, k)-modules M are subquotie...

The third part of J.-P. Serre’s book Linear Representations of Finite Groups discusses of modular representations, i.e., representations of finite groups over a field of nonzero characteristic. The aim of this work is to complement Serre’s work by introducing the reader to some noncommutative algebra used in the study of modular representations, and in particular to the theory of semisimple and...

Let C be a finite braided multitensor category. Then the end B=∫X∈CX⊗X⁎ is natural Hopf algebra in C. We show that Drinfeld center of isomorphic to category left B-comodules C, and decomposition B into direct sum indecomposable C-subcoalgebras leads B-ComodC C-module subcategories. As an application, we present explicit characterization structure irreducible Yetter-Drinfeld modules over semisim...

We consider bases for the cohomology space of regular semisimple Hessenberg varieties, consisting classes that naturally arise from Białynicki-Birula decomposition varieties. give an explicit combinatorial description support each class, which enables us to compute symmetric group actions on in our bases. then successfully apply results permutohedral varieties explicitly write down class and co...

Kleshchev has recently [7] classi®ed those modules for the symmetric group which have semisimple restriction to any Young subgroup. We determine Ext K Sd …D ;D m† where D l and D m are K Sd -modules of this type, called completely splittable. As a corollary of this and recent work of Kleshchev and Nakano, we can determine ExtGL n…K†……L…l†;L…m†† for certain simple GLn…K†-modules L…l† and L…m†. 1...