Representations of Quantum Conjugacy Classes of Non-Exceptional Quantum Groups
thesisposted on 21.12.2016, 16:00 by Thomas Stephen Ashton
Let G be a complex non-exceptional simple algebraic group and g its Lie algebra. With every point x of the maximal torus T ʗ G we associate a highest weight module Mx over the Drinfeld-Jimbo quantum group Uq(g) and an equivariant quantization of the conjugacy class of x by operators in End(Mx). These equivariant quantizations are isomorphic for x lying on the same orbit of the Weyl group, and Mx support different exact representations of the same quantum conjugacy class. This recovers all quantizations of conjugacy classes constructed before, via special x, and also completes the family of conjugacy classes by constructing an equivariant quantization of “borderline" Levi conjugacy classes of the complex orthogonal group SO(N), whose stabilizer contains a Cartesian factor SO(2) SO(P), 1 6 P < N, P Ξ N mod 2. To achieve this, generators of the Mickelsson algebra Zq(g; g’), where g’ ʗ g is the Lie subalgebra of rank rkg’ = rkg-1 of the same type, were explicitly constructed in terms of Chevalley generators via the R-matrix of Uq(g).