by

DOI: 10.1023/A:1017468526647
Print publication date: 3/1/2002
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by Joseph, Anthony; Todorić, Dragana

Let be a semisimple or affine Lie algebra and U

_q
() its quantized enveloping algebra. Extending earlier work, the KPRV determinant for an admissible integrable U

_q
() module V relative to a parabolic subalgebra ⊂ is defined and shown to be nonzero. These determinants had previously been evaluated for semisimple and a Borel subalgebra. The present results can be used to extend this to affine as will be shown in a subsequent publication.For a parabolic subalgebra the evaluation of these determinants is much more difficult. For appropriate overalgebras of the primitive quotients of the enveloping algebra U() defined by one-dimensional representations of , these determinants had been calculated for semisimple. However the quantum case is interesting because it is unnecessary to pass to overalgebras and besides for U(): affine, it is not even clear how these determinants should be defined. Here for semisimple, the degrees of the determinants are computed and shown to depend on being the same type of functions as in the enveloping algebra case; yet in a different fashion. Some special cases (in type A
₄) are computed explicity. Here, as in the Borel case, the determinants take a remarkably simple form and notably can be expressed as a product of ‘linear’ factors. However compared to the enveloping algebra case one finds additional factors corresponding to what are called quantum zeros and whose origin remains unknown.

DOI: 10.1023/A:1014410819083
Print publication date: 3/1/2002
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by Atkinson, M. D.; Pfeiffer, G.; van Willigenburg, S. J.

The concept of descent algebras over a field of characteristic zero is extended to define descent algebras over a field of prime characteristic. Some basic algebraic structure of the latter, including its radical and irreducible modules, is then determined. The decomposition matrix of the descent algebras of Coxeter group types A, B, and D are calculated, and used to derive a description of the decomposition matrix of an arbitrary descent algebra. The Cartan matrix of a variety of descent algebras over a finite field is then obtained.

DOI: 10.1023/A:1014413413572
Print publication date: 3/1/2002
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by Mundkur, Arun

This article provides an introduction to A. Bak’s theory of group-valued functors on categories with structure and dimension and applies this theory to the algebraic K-theory functors nonstable K
₁ and K
₂.

DOI: 10.1023/A:1014416120970
Print publication date: 3/1/2002
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