Nikolai Iorgov

Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine
14b Metrologichna Str., 03143, Kyiv, Ukraine
e-mail: mmtpitp@bitp.kiev.ua

Spectra of Casimir operators of q-deformed algebras U'q(son) related to quantum gravity

Abstract:
It is known that the nonstandard q-deformed algebras U'q(son) appear as auxiliary algebras in deriving the algebra of observables in 2+1 quantum gravity with 2D space of genus g. In order to obtain the algebra of observables, the q-deformed algebra U'q(son) should be quotiented by some ideal generated by (combinations of) Casimir elements of this algebra. This fact, along with others, motivates the study of Casimir elements of U'q(son). We give in explicit form the eigenvalues of Casimir operators in irreducible representations of classical and nonclassical types when q is not a root of unity.
Other part of the talk is devoted to Fairlie-Odesskii algebra U'q(so3) which is known to possess cyclic irreducible representations when q is a root of unity.
The eigenvalues of Casimir operators acting in the spaces of these representations are presented in explicit form. These eigenvalues are algebraically dependent. The relation between them is given in terms of Chebyshev polynomials of the first kind.