Extensions of Modules over Schur Algebras, Symmetric Groups and Hecke Algebras

We study the relation between the cohomology of general linear and symmetric groups and their respective quantizations, using Schur algebras and standard homological techniques to build appropriate spectral sequences. As our methods fit inside a much more general context within the theory of finite-dimensional algebras, we develop our results first in that general setting, and then specialize to the above situations. From this we obtain new proofs of several known results in modular representation theory of symmetric groups. Moreover, we reduce certain questions about computing extensions for symmetric groups and Hecke algebras to questions about extensions for general linear groups and their quantizations.     

Hecke Algebras of Group Extensions

Abstract

We describe the Hecke algebra ?(Γ,Γ₀) of a Hecke pair (Γ,Γ₀) in terms of the Hecke pair (N,Γ₀) where N is a normal subgroup of Γ containing Γ₀. To do this, we introduce twisted crossed products of unital *-algebras by semigroups. Then, provided a certain semigroup S ? Γ/N satisfies S ^?1 S = Γ/N, we show that ? (Γ,Γ₀) is the twisted crossed product of ? (N,Γ₀) by S. This generalizes a recent theorem of Laca and Larsen about Hecke algebras of semidirect products.     

Modules for Yokonuma-type Hecke Algebras

This paper describes the module categories for a family of generic Hecke algebras, called Yokonuma-type Hecke algebras. Yokonuma-type Hecke algebras specialize both to the group algebras of the complex reflection groups G(r,1,n) and to the convolution algebras of (B \(^{\prime }\),B \(^{\prime }\))-double cosets in the group algebras of finite general linear groups, for certain subgroups B \(^{\prime }\) consisting of upper triangular matrices. In particular, complete sets of inequivalent, irreducible modules for semisimple specializations of Yokonuma-type Hecke algebras are constructed.     

Parabolically Induced Representations of Graded Hecke Algebras

We study the representation theory of graded Hecke algebras, starting from scratch and focusing on representations that are obtained by induction from a discrete series representation of a parabolic subalgebra. We determine all intertwining operators between such parabolically induced representations, and use them to parametrize the irreducible representations. Finally we describe the spectrum of a graded Hecke algebra as a topological space.     

Whittaker Modules for Graded Lie Algebras

In this paper, we study Whittaker modules for graded Lie algebras over ℂ. We define Whittaker modules for a class of graded Lie algebras and obtain a bijective correspondence between the set of isomorphism classes of Whittaker modules and the set of ideals of a polynomial ring, parallel to a result from the classical setting and the case of the Virasoro algebra. As a consequence of this, we obtain a classification of simple Whittaker modules for such algebras. Also, we discuss some concrete algebras as examples.     

On Characteristic Modules of Graded Quasi-hereditary Algebras

Abstract

Let G be a group and A a G-graded quasi-hereditary algebra. Then its characteristic module is proved to be G-gradable, i.e., it is isomorphic to a G-graded module as A-modules. This implies that the Ringel dual A′ of A admits a canonical G-grading which extends to the graded situation the typical equivalence between Δ-good and ?-good modules of A and A′, respectively. It follows some consequences: the derived category of finitely generated G-graded A-modules is equivalent to the derived category of finitely generated G-graded A'-modules; if G is finite, then the Ringel dual of the smash product A#G* is isomorphic to the smash product A'#G* of A' with G.     

Extensions of Modules over Hopf Algebras Arising from Lie Algebras of Cartan Type

In this paper we prove that there are no self-extensions of simple modules over restricted Lie algebras of Cartan type. The proof given by Andersen for classical Lie algebras not only uses the representation theory of the Lie algebra, but also representations of the corresponding reductive algebraic group. The proof presented in the paper follows in the same spirit by using the construction of a infinite-dimensional Hopf algebra D(G) u( ) containing u( ) as a normal Hopf subalgebra, and the representation theory of this algebra developed in our previous work. Finite-dimensional hyperalgebra analogs D(G _r ) u( ) have also been constructed, and the results are stated in this setting.     

Endomorphism Algebras of Some Modules for Schur Algebras and Representation Dimension

We consider the representation dimension, for fixed n ≥ 2, of ordinary and quantised Schur algebras S(n, r) over a field k. For k of positive characteristic p we give a lower bound valid for all p. We also give an upper bound in the quantum case, when k has characteristic 0.     

Irreducible Morphisms Between Modules over a Repetitive Algebras

We describe the irreducible morphisms in the category of modules over a repetitive algebra. We find three special canonical forms: The first canonical form happens when all the component morphisms are split monomorphisms, the second when all the component morphisms are split epimorphisms and the third when there is exactly one irreducible component map. Also, we obtain the same result for the irreducible homomorphisms in the stable category of modules over a repetitive algebra.     

Involutions of Double Affine Hecke Algebras

We prove the existence of an involution on double affine Hecke algebras. This involution plays a central role in the theory of Macdonald polynomials, being responsible for a Fourier type duality and allowing the construction of affine intertwiners.     

Hochschild Homology of Tame Hecke Algebras

Let A be a tame Hecke algebra of type A. A new minimal projective bimodule resolution for A is constructed and the dimensions of all the Hochschild homology groups and cyclic homology groups are calculated explicitly.     

Rees Algebras of Modules

We study Rees algebras of modules within a fairly general framework.We introduce an approach through the notion of Bourbaki idealsthat allows the use of deformation theory. One can talk aboutthe (essentially unique) generic Bourbaki ideal I(E) of a moduleE which, in many situations, allows one to reduce the natureof the Rees algebra of E to that of its Bourbaki ideal I(E).Properties such as Cohen–Macaulayness, normality and beingof linear type are viewed from this perspective. The known numericalinvariants, such as the analytic spread, the reduction numberand the analytic deviation, of an ideal and its associated algebrasare considered in the case of modules. Corresponding notionsof complete intersection, almost complete intersection and equimultiplemodules are examined in some detail. Special consideration isgiven to certain modules which are fairly ubiquitous becauseinteresting vector bundles appear in this way. For these modulesone is able to estimate the reduction number and other invariantsin terms of the Buchsbaum–Rim multiplicity. 2000 MathematicsSubject Classification 13A30 (primary), 13H10, 13B21 (secondary)     

N-Crossed Extensions of Crossed Modules

Abstract

We obtain cohomological δ-functors and a five term exact sequence associated to an extension of crossed modules which yields Eilenberg-MacLane cohomology theory and the classic five term exact sequence associated to an extension of groups, when we consider the particular situation of groups.     

Generic Extensions of Prinjective Modules

Let be an algebraically closed field and let be a finite partially ordered set of finite prinjective type. We study generic extensions of prinjective modules over the incidence -algebra of . We prove that there exist generic extensions of prinjective -modules and describe properties of the monoid of generic extensions.Partially supported by Polish KBN Grant 5 P0 3A 015 21.     

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.     

Ore Extensions of Hopf Algebras

In this paper, Ore extensions in the class of Hopf algebras are studied. The classification theorem enables one to describe the Hopf--Ore extensions for the group algebras, for the algebras and , and for the quantum ax + b group.