by Cavallaro, Stefano

Let
$${\mathcal{A}}$$
be an Abelian unital C
^*-algebra and let
$$\hat {\mathcal{A}}$$
denote its Gelfand spectrum. We give some necessary and sufficient conditions for a nondegenerate representation of
$${\mathcal{A}}$$
to be unitarily equivalent to a representation in which the elements of
$${\mathcal{A}}$$
act multiplicatively, by their Gelfand transforms, on a space L
²(
$$\hat {\mathcal{A}}$$
,μ), where μ is a positive measure on the Baire sets of
$$\hat {\mathcal{A}}$$
. We also compare these conditions with the multiplicity-free property of a representation.

DOI: 10.1023/A:1009941513419
Print publication date: 6/1/2000
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by Jakobsen, Hans Plesner; Zhang, Hechun

We investigate the algebra F

_q
(N) introduced by Faddeev, Reshetikhin and Takhadjian. In the case where q is a primitive root of unity, the degree, the center, and the set of irreducible representations are found. The Poisson structure is determined and the De Concini–Kac–Procesi Conjecture is proved for this case.

DOI: 10.1023/A:1009937429349
Print publication date: 6/1/2000
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by Joseph, A.

This paper studies a class of simple integrable modules for an affine Lie algebras which are closely related to the finite-dimensional modules studied by V. Chari and A. Pressley, except that the Euler element is assumed to act. They are infinite-dimensional; but are shown to have finite-dimensional weight spaces. It is conjectured that any simple integrable module with a zero weight space belongs to this class and their classification is given. The main interest in studying such modules is that they may occur in the endomorphism rings of highest weight modules whilst those of Chari and Pressley in general do not. Their character theory is also more complicated.

DOI: 10.1023/A:1009985328440
Print publication date: 6/1/2000
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by Groza, V. A.; Iorgov, N. Z.; Klimyk, A. U.

The main aim of the paper is to study infinite-dimensional representations of the real form U

_q
(u
_{n, 1}) of the quantized universal enveloping algebra U

_q
(gl
_{n + 1}). We investigate the principal series of representations of U

_q
(u
_{n, 1}) and calculate the intertwining operators for pairs of these representations. Some of the principal series representations are reducible. The structure of these representations is determined. Then we classify irreducible representations of U

_q
(u
_{n, 1}) obtained from irreducible and reducible principal series representations. All *-representations in this set of irreducible representations are separated. Unlike the classical case, the algebra U

_q
(u
_{n, 1}) has finite-dimensional irreducible *-representations.

DOI: 10.1023/A:1009906111602
Print publication date: 6/1/2000
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