On Fock Space Representations of quantized Enveloping Algebras related to Non-Commutative Differential Geometry

TL;DR

This paper constructs explicit Fock space representations of quantized enveloping algebras using geometric methods involving differential operators on q-deformed flag manifolds, linking algebraic and geometric perspectives.

Contribution

It introduces a geometric construction of Fock space representations of quantum groups via differential operators on q-deformed flag manifolds, based on the Gauss decomposition.

Findings

Explicit construction of Fock space representations

Representation of algebra elements as differential operators

Connection between algebraic and geometric structures

Abstract

In this paper we construct explicitly natural (from the geometrical point of view) Fock space representations (contragradient Verma modules) of the quantized enveloping algebras. In order to do so, we start from the Gauss decomposition of the quantum group and introduce the differential operators on the corresponding $q$-deformed flag manifold (asuumed as a left comodule for the quantum group) by a projection to it of the right action of the quantized enveloping algebra on the quantum group. Finally, we express the representatives of the elements of the quantized enveloping algebra corresponding to the left-invariant vector fields on the quantum group as first-order differential operators on the $q$-deformed flag manifold.