Anatoly Klimyk

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.