by Noels, Jeroen

In a previous article we proposed an algebraic setting in which to perform harmonic analysis on noncompact, nondiscrete quantum groups and in particular, on quantum E(2). In the present paper we shall explicitly construct Fourier transforms between quantum E(2) and its Pontryagin dual, involving Hahn—Exton q-Bessel functions as kernel, prove Plancherel and inversion formulas, etc. We also develop a theory of q-Hankel transformation of entire functions, based on the definition proposed by Koornwinder and Swarttouw.

DOI: 10.1023/B:ALGE.0000026845.01180.1d
Print publication date: 5/1/2004
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by Natale, Sonia

Let p and q be distinct prime numbers. We prove a result on the existence of nontrivial group-like elements in a certain class of semisimple Hopf algebras of dimension pq

^r
. We conclude the classification of semisimple Hopf algebras A of dimension pq
² over an algebraically closed field k of characteristic zero, such that both A and A
^* are of Frobenius type. We also complete the classification of semisimple Hopf algebras of dimension pq
²<100.

DOI: 10.1023/B:ALGE.0000026827.21137.fa
Print publication date: 5/1/2004
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by Hertweck, Martin; Nebe, Gabriele

For a finite group G, the group Outcent(Z

_p

G) of outer central automorphisms of Z

_p

G only depends on the Morita equivalence class of Z

_p

G, which allows reduction to a basic order for its calculation. If the group ring is strongly related to a graduated order, it is often possible to give an explicit description of the basic order. In this paper, we show that Outcent(B)=1 for a block B of Z

_p

G with cyclic defect group. We also prove that Outcent(B₀⁽³⁾(A
₆))= 1 for the principal block B₀⁽³⁾(A
₆) of Z
₃
A
₆; this allows us to verify a conjecture of Zassenhaus for the perfect group of order 1080.

DOI: 10.1023/B:ALGE.0000026826.81439.3f
Print publication date: 5/1/2004
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by An, Jianbei; O’Brien, E. A.

We classify the radical subgroups and chains of the Conway simple group Co₁ and then verify the Alperin weight conjecture and the Dade final conjecture for this group.

DOI: 10.1023/B:ALGE.0000026809.78997.19
Print publication date: 5/1/2004
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by Nenciu, Adriana

We study the quasitriangular structures for a family of pointed Hopf algebras which is big enough to include Taft’s Hopf algebras H

_n

², Radford’s Hopf algebras H

_N,n,q,ν and E(n). We give necessary and sufficient conditions for the Hopf algebras in our family to be quasitriangular. For the case when they are, we determine completely all the quasitriangular structures. Also, we determine the ribbon elements of the quasitriangular Hopf algebras and the quasi-ribbon elements of their Drinfel’d double.

DOI: 10.1023/B:ALGE.0000026785.03997.60
Print publication date: 5/1/2004
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