\bf A Borel-Weil-Bott approach to representations of $\slq$

TL;DR

This paper introduces a concrete realization of the quantum group sl_q using finite difference operators interpreted as derivations, leading to a non-commutative geometric model of the projective line.

Contribution

It presents a novel realization of sl_q via finite difference operators and interprets them as derivations on a graded algebra, connecting quantum groups with non-commutative geometry.

Findings

Realization of sl_q using finite difference operators

Interpretation of operators as derivations in non-commutative algebra

Construction of a non-commutative version of the projective line

Abstract

We use a quite concrete and simple realization of $\slq$ involving finite difference operators. We interpret them as derivations (in the non-commutative sense) on a suitable graded algebra, which gives rise to the double of the projective line as the non commutative version of the standard homogeneous space.