by Huisgen-Zimmermann, B.; SmalØ, S. O.

Given a finite-dimensional algebra Λ, we show that a frequently satisfied finiteness condition for the category
$$P^\infty$$
Λ-mod) of all finitely generated (left) Λ-modules of finite projective dimension,namely contravariant finiteness of
$$P^\infty$$
(Λ-mod) in Λ-mod, forces arbitrary modules of finite projective dimension to be direct limits of objects in
$$P^\infty$$
(Λ-mod). Among numerous applications, this yields an encompassing sufficient condition for the validity of the first finitistic dimensionconjecture, that is, for the little finitistic dimension of Λ to coincide with the big (this is well known to fail overfinite-dimensional algebras in general).

DOI: 10.1023/A:1009948721602
Print publication date: 6/1/1998
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by Hanaki, Akihide; Koshitani, Shigeo

Let p be a prime, k an algebraically closed field ofcharacteristic p, and G a finite group with a Sylowp-Subgroup P. In this paper, we consider the property thatNG(P)/P is Abelian. We provide somenecessary or sufficient conditions NG(P)/P to be Abelian in term of thestructure of the group algebra kG as a k-algebra, in casethat G is p-nilpotent or of p-length 1.

DOI: 10.1023/A:1009937601589
Print publication date: 6/1/1998
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by Futorny, Viatcheslav M.; Melville, Duncan J.

We construct quantum analogues of a class of generalized Verma modulesinduced from nonsolvable parabolic subalgebras of simple Lie algebras. Weshow that these quantum modules are true deformations of the underlyingclassical modules in the sense that the weight-space decomposition ispreserved.

DOI: 10.1023/A:1009945008915
Print publication date: 6/1/1998
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by Neidhardt, Wayne

We prove the following result for a not necessarily symmetrizable Kac–Moody algebra: Let x,y ∈ W with x ≥ y, and let λ ∈ P⁺. If n=l(x)-l(y), then Ext _C(λ)

ⁿ
(M(x·λ),L(y·λ))=1.

DOI: 10.1023/A:1009933409824
Print publication date: 6/1/1998
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by Donkin, Stephen

This is the first part of our study of the existence ofAuslander–Reiten sequences of group representations. In this part weconsider representations of group schemes in characteristic 0; in Part II weconsider representations of group schemes in characteristic p; andin Part III we give applications to representations of groups and Liealgebras.

DOI: 10.1023/A:1009968402535
Print publication date: 6/1/1998
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by Robinson, Geoffrey R.

In this paper, we prove that a projective generalization of theKnörr–Robinson formulation of Alperin’s conjecture holds ifthe ‘ordinary’ form holds for a certain quotient group.

DOI: 10.1023/A:1009940803444
Print publication date: 6/1/1998
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