Simple Compact Quantum Groups I

Quantum Galois theory for compact Lie groups

We establish a quantum Galois correspondence for compact Lie groups of automorphisms acting on a simple vertex operator algebra.

Reiter’s Properties for the Actions of Locally Compact Quantum Goups on von Neumann Algebras

Quadratic independence of coordinate functions of certain homogeneous spaces and action of compact quantum groups

Let G be one of the classical compact, simple, centre-less, connected Lie groups of rank n with a maximal torus T , the Lie algebra G and let {Ei , Fi , Hi , i = 1, . . . , n} be the standard set of generators corresponding to a basis of the root system. Consider the adjoint-orbit space M = {Adg(H1), g ∈ G}, identified with the homogeneous space G/L where L = {g ∈ G : Adg(H1) = H1}. We prove th...

Locally compact quantum groups in the universal setting

In this paper we associate to every reduced C∗-algebraic quantum group (A,∆) (as defined in [18]) a universal C∗-algebraic quantum group (Au,∆u). We fine tune a proof of Kirchberg to show that every ∗-representation of a modified L-space is generated by a unitary corepresentation. By taking the universal enveloping C∗-algebra of a dense sub ∗-algebra of A we arrive at the C∗-algebra Au. We show...

Minimal Uncertainty States for Quantum Groups

The problem of how to obtain quasi-classical states for quantum groups is examined. A measure of quantum indeterminacy is proposed, which involves expectation values of some natural quantum group operators. It is shown that within any finite dimensional irreducible representation, the highest weight vector and those unitarily related to it are the quasi-classical states. Quantum groups [1] have...