an:00859669
Zbl 0868.33010
Kirillov, A. A. jun.; Etingof, P. I.
A unified representation-theoretic approach to special functions
EN
Funct. Anal. Appl. 28, No. 1, 73-76 (1994); translation from Funkts. Anal. Prilozh. 28, No. 1, 91-94 (1994).
00025383
1994
j
33C80 16W30 17B37 81R30
Hopf algebra; Clebsch-Gordan coefficients
The paper provides a new method of obtaining classes of special functions by means of the group-theoretic approach.
Let \(\mathcal H\) be a Hopf algebra and \(H\subset{\mathcal H}\) a subgroup of the group of invertible elements. Let \(U\), \(V\), \(W\) be irreducible \({\mathcal H}\)-modules and \(\varphi:V\to W\otimes U\) an intertwining operator for \(\mathcal H\).
The functions \(f_{vw\varphi}(h)= \langle w,\varphi hv\rangle\in U\) (\(h\in H\), \(v\in V\), \(w\in W^*\)) are called the \(U\)-valued matrix elements. In case of \(V=W\) the function \(\chi_\varphi(h)= \text{Tr}|_V(\varphi h)\) is called the \(U\)-valued character.
The paper shows that in particular cases the above construction leads to (1) the usual matrix elements and character of group representations; (2) the Clebsch-Gordan coefficients; (3) the matrix elements of intertwining operators corresponding to representations of quantum affine Lie groups.
The authors give the generalized Peter-Weyl theorem for the vector-valued elements and characters of finite groups and compact Lie groups.
The most important applications correspond to \(\mathcal H\) equal to the convolution algebra of distributions on a Lie group \(G\) and to \(\mathcal H\) equal to the quantum affine algebra, \(U_q(\widehat{\mathfrak g})\) for \(\mathfrak g\) a simple Lie algebra. Three particular cases are worked out with details.
A.Wawrzynczyk (MR 95h:33010)