by Malinin, D. A.

For a given field F of characteristic 0 we consider a normal extension E/F of finite degree d and finite Abelian subgroups G⊂GL

_n
(E) of a given exponent t. We assume that G is stable under the natural action of the Galois group of E/F and consider the fields E=F(G) that are obtained via adjoining all matrix coefficients of all matrices g∈G to F. It is proved that under some reasonable restrictions for n, any E can be realized as F(G), while if all coefficients of matrices in G are algebraic integers, there are only finitely many fields E=F(G) for prescribed integers n and t or prescribed n and d.

DOI: 10.1023/A:1023254607196
Print publication date: 5/1/2003
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by Bruns, Winfried; Gubeladze, Joseph

We investigate the minimal number of generators μ and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result is that for every bound C there exist, up to isomorphism, only finitely many divisorial ideals I such that μ(I)≤C. It follows that there exist only finitely many Cohen–Macaulay divisor classes. Moreover, we determine the minimal depth of all divisorial ideals and the behaviour of μ and depth in ‘arithmetic progressions’ in the divisor class group.The results are generalized to more general systems of linear inequalities whose homogeneous versions define the semigroup in a not necessarily irredundant way. The ideals arising this way can also be considered as defined by the nonnegative solutions of an inhomogeneous system of linear diophantine equations.We also give a more ring-theoretic approach to the theorem on minimal number of generators of divisorial ideals: it turns out to be a special instance of a theorem on the growth of multigraded Hilbert functions.

DOI: 10.1023/A:1023295114933
Print publication date: 5/1/2003
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by Drensky, Vesselin

We obtain defining relations of the algebra of invariants of the classical subgroups of GL
₂(C) acting by simultaneous conjugation on m-tuples of 2×2 complex matrices. The sets of defining relations look uniformly for all m≥2 and are derived by translation of classical results on invariant theory of orthogonal groups in the language of 2×2 matrix invariants, combined with arguments of representation theory of the general linear group GL

_m
(C) and ideas coming from the theory of algebras with polynomial identities.

DOI: 10.1023/A:1023266314025
Print publication date: 5/1/2003
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by Hajac, Piotr M.; Matthes, Rainer; Szymański, Wojciech

We define the C
^*-algebra of quantum real projective space R
P

_q

², classify its irreducible representations, and compute its K-theory. We also show that the q-disc of Klimek and Lesniewski can be obtained as a non-Galois Z
₂-quotient of the equator Podleś quantum sphere. On the way, we provide the Cartesian coordinates for all Podleś quantum spheres and determine an explicit form of isomorphisms between the C
^*-algebras of the equilateral spheres and the C
^*-algebra of the equator one.

DOI: 10.1023/A:1023288309786
Print publication date: 5/1/2003
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by Greenstein, Jacob

After V. Chari and A. Pressley, a simple integrable module with finite-dimensional weight spaces over an affine Lie algebra is either a standard module (highest or lowest weight), in which case its formal character is given by the famous Weyl–Kac formula, or a subquotient of a tensor product of loop modules. In this paper we compute formal characters of generic simple integrable modules of the latter type.

DOI: 10.1023/A:1023217107637
Print publication date: 5/1/2003
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