Coherent States for Quantum Compact Groups

TL;DR

This paper develops a framework for coherent states on quantum compact groups, linking representation theory with non-commutative geometry, and generalizing classical geometric structures to the quantum setting.

Contribution

It introduces coherent states for all simple quantum compact groups, providing explicit holomorphic coordinates and relations, and extends the Borel--Weil construction to the quantum context.

Findings

Explicit holomorphic coordinates on quantum dressing orbits

Derivation of commutation relations in R-matrix form

Extension of Borel--Weil construction to quantum groups

Abstract

Coherent states are introduced and their properties are discussed for all simple quantum compact groups. The multiplicative form of the canonical element for the quantum double is used to introduce the holomorphic coordinates on a general quantum dressing orbit and interpret the coherent state as a holomorphic function on this orbit with values in the carrier Hilbert space of an irreducible representation of the corresponding quantized enveloping algebra. Using Gauss decomposition, the commutation relations for the holomorphic coordinates on the dressing orbit are derived explicitly and given in a compact R--matrix formulation (generalizing this way the $q$--deformed Grassmann and flag manifolds). The antiholomorphic realization of the irreducible representations of a compact quantum group (the analogue of the Borel--Weil construction) are described using the concept of coherent state. The relation between representation theory and non--commutative differential geometry is suggested.}