Bogolyubov Institute for Theoretical Physics of the National Academy
of Sciences of Ukraine
14b Metrologichna Str., 03143, Kyiv, Ukraine
e-mail: aklimyk@bitp.kiev.ua
On classification of irreducible representations of the q-deformed algebra U'q(son) related to quantum gravity
Abstract:
The universal enveloping algebra U(son) of the Lie
algebra son of the rotation group SO(n) has two
different structures. The first one is related to roots and root elements.
Quantization (q-deformation) of this structure leads to Drinfeld-Jimbo
quantized universal enveloping algebra Uq(son).
The second structure is related to the realization of U(son)
by skew-symmetric matries. A q-deformation of this structure leads
to the q-deformed algebra U'q(son) which differs
from the Drinfeld-Jimbo algebra Uq(son). It
turns out that the algebra U'q(son) is related
to quantum gravity, algebraic geometry, the theory of q-harmonic
polynomials, etc. The aim of the talk is to expose the status of classification
of irreducible finite dimensional representations of this algebra when
q is not a root of unity and when q is a root of unity.