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.     

Valuation Extensions of Filtered and Graded Algebras

In this note we relate the valuations of the algebras appearing in the noncommutative geometry of quantized algebras to properties of sublattices in some vector spaces. We consider the case of algebras with PBW-bases and prove that under some mild assumptions, the valuations of the ground field extend to a noncommutative valuation. Later we introduce the notion of F-reductor and graded reductor and reduce the problem of finding an extending noncommutative valuation to finding a reductor in an associated graded ring having a domain for its reduction.     

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.     

Generalized Graded Hecke Algebras of Types B and D

For a root system of type B we study an algebra similar to a graded Hecke algebra, isomorphic to a subalgebra of the rational Cherednik algebra. We introduce principal series modules over it and prove an irreducibility criterion for these modules. We deduce similar results for an algebra associated to a root system of type D.     

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.     

Leibniz Central Extensions on Some Infinite-Dimensional Lie Algebras

Abstract

In this paper, all one-dimensional Leibniz central extensions on the algebras of differential operators over C[t, t ^?1] and C((t)), as well as on the quantum 2-torus, the Virasoro-like algebra and its q-analog are studied. We determine all nontrivial Leibniz 2-cocycles on these infinite dimensional Lie algebras.     

某些$W$-型李代数的Verma模

本文我们研究了两类$W$-型李代数$g(\lambda)$的Verma模的结构. 在一定条件下,我们决定了这些Verma模的可约性及相应的奇异向量.     

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.     

分次环的分次Excellent扩张

本文引进了分次环的分次Excellent扩张概念,设S=⊕_(g∈G)S_g是R=⊕_(g∈G)R_g的分次Excellent扩张,证明了S是分次右V-环当且仅当R是分次右V-环,S是分次PS-环当且仅当R是分次PS-环,S是分次Von Neumann正则环当且仅当R是分次Von Neumann正则环。     

Some Results about Projective Modules over Monoid Algebras

(1) Let R be a 1-dimensional commutative Noetherian anodal ring with finite seminormalization and M a commutative cancellative torsion-free monoid. Let P be a projective R[M]-module of rank r. Then P ? ∧^rP ⊕ R[M]^r?1.

(2) Murthy and Pedrini [11 Murthy, M. P., Pedrini, C. (1973). K₀ and K₁ of polynomial rings. In: Algebraic K-Theory, II: “Classical” Algebraic K-Theory and Connections with Arithmetic (Proc. Conf., Seattle Res. Center, Battelle Memorial Inst., Seattle, Wash., 1972). Lecture Notes in Math., Vol. 342. Berlin: Springer, pp. 109–121.[Crossref] , [Google Scholar]] proved K₀ homotopy invariance of polynomial extension of some affine normal surfaces. We extend this result to a monoid extension (see 1.5).