3 research outputs found
On representation theory of quantum groups at roots of unity
Irreducible representations of quantum groups (in Woronowicz'
approach) were classified in J.Wang, B.Parshall, Memoirs AMS 439 in the~case of
being an~odd root of unity. Here we find the~irreducible representations
for all roots of unity (also of an~even degree), as well as describe
"the~diagonal part" of tensor product of any two irreducible representations.
An~example of not completely reducible representation is given. Non--existence
of Haar functional is proved. The~corresponding representations of universal
enveloping algebras of Jimbo and Lusztig are provided. We also recall the~case
of general~. Our computations are done in explicit way.Comment: 31 pages, Section 2.7 added and other minor change
Fusion Rules of the Lowest Weight Representations of osp_q(1|2) at Roots of Unity: Polynomial Realization and Degeneration at Roots of Unity
The degeneracy of the lowest weight representations of the quantum
superalgebra and their tensor products at exceptional values of
%when deformation parameter takes exceptional values is studied. The main
features of the structures of the finite dimensional lowest weight
representations and their fusion rules are illustrated using realization of
group generators as finite-difference operators acting in the space of the
polynomials. The complete fusion rules for the decompositions of the tensor
products at roots of unity are presented. The appearance of indecomposable
representations in the fusions is described using Clebsh-Gordan coefficients
derived for general values of and at roots of unity.Comment: 28 pages, 3 figures; completed (section 5, section 6); version
published in JP
Duality for Exotic Bialgebras
In the classification of Hietarinta, three triangular
-matrices lead, via the FRT formalism, to matrix bialgebras which are not
deformations of the trivial one. In this paper, we find the bialgebras which
are in duality with these three exotic matrix bialgebras. We note that the
duality of FRT is not sufficient for the construction of the bialgebras
in duality. We find also the quantum planes corresponding to these bialgebras
both by the Wess-Zumino R-matrix method and by Manin's method.Comment: 25 pages, LaTeX2e, using packages: cite, amsfonts, amsmath, subeq