Unitary Representations of Noncompact Quantum Groups at Roots of Unity

Abstract

Noncompact forms of the Drinfeld-Jimbo quantum groups U f in q (g) withare studied at roots of unity. This covers g = so(n, 2p), su(n, p), so * (2l), sp(n, p), sp(l, R), and exceptional cases. Finite-dimensional unitary representations are found for all these forms, for even roots of unity. Their classical symmetry induced by the Frobenius-map is determined, and the meaning of the extra quasiclassical generators appearing at even roots of unity is clarified. The unitary highest weight modules of the classical case are recovered in the limit q → 1.

Quantum anti-de Sitter space and sphere at roots of unity

“…In fact, U res q "contains" a corresponding classical Lie algebra as a quotient. This is the essence of a remarkable result of Lusztig [23], and can be made explicit as follows ( [37], Theorem 4.2):…”

“…This is not hard to prove, see [5] or [37]. Moreover, the generators (2.36) essentially act on the second factor in (2.37) and U f in q on the first, but with a certain "twisting" [37].…”

“…There are essentially 2 types of star structures 3 on U q [37], the first of the form 4) and the second of the form…”

“…The compatibility conditions with the coproduct are different for real q and q a phase, which will show up in different reality structures of the associated quantum spaces. It is known that there exist finite-dimensional unitary representations of the first type (3.4) if q is a root of unity [38,37], which have the desired properties including a high-energy cutoff in the AdS case. For q ∈ R, unitary representations of noncompact forms also exist, but they have no cutoff.…”

“…Moreover, it will turn out that the definition of AdS D q implies a number of additional, unexpected features. They include the appearance of an additional undeformed symmetry group SO(D + 1) if D is odd and Sp(D) if D is even, which are in some sense spontaneously broken [37]. Moreover, it turns out that the quantum spaces are obtained most naturally as product of the quantum AdS space (or sphere) with a classical sphere.…”