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.