Quantized algebras of functions on homogeneous spaces with Poisson stabilizers

TL;DR

This paper studies the quantization of function algebras on homogeneous spaces with Poisson structures, classifies their representations, describes their topology, and establishes their continuous deformation and KK-equivalence to classical algebras.

Contribution

It provides a detailed classification of irreducible representations and a continuous field structure for quantized function algebras on homogeneous spaces with Poisson stabilizers.

Findings

Classified irreducible representations of C(G_q/K_q).

Described the topology of the spectrum via symplectic leaves.

Established continuous deformation from classical to quantum algebras.

Abstract

Let G be a simply connected semisimple compact Lie group with standard Poisson structure, K a closed Poisson-Lie subgroup, 0<q<1. We study a quantization C(G_q/K_q) of the algebra of continuous functions on G/K. Using results of Soibelman and Dijkhuizen-Stokman we classify the irreducible representations of C(G_q/K_q) and obtain a composition series for C(G_q/K_q). We describe closures of the symplectic leaves of G/K refining the well-known description in the case of flag manifolds in terms of the Bruhat order. We then show that the same rules describe the topology on the spectrum of C(G_q/K_q). Next we show that the family of C*-algebras C(G_q/K_q), 0<q\le1, has a canonical structure of a continuous field of C*-algebras and provides a strict deformation quantization of the Poisson algebra \C[G/K]. Finally, extending a result of Nagy, we show that C(G_q/K_q) is canonically KK-equivalent to C(G/K).