Braided Symmetric Algebras of Simple $U_q(sl_2)$-Modules and Their Geometry

TL;DR

This paper develops decomposition formulas for braided symmetric powers of simple modules over quantum groups, revealing their geometric structures as non-commutative curves and surfaces, and proposing a new perspective on quantum non-commutative geometry.

Contribution

It introduces quantum analogues of classical symmetric powers and explores their geometric properties, linking algebraic decompositions to non-commutative geometric structures.

Findings

Braided symmetric powers decompose according to specific formulas.

Point modules form non-commutative curves and surfaces.

Proposes non-commutative geometry as a deformation of classical limits.

Abstract

In the present paper we prove decomposition formulae for the braided symmetric powers of simple modules over the quantized enveloping algebra $U_q(sl_2)$; natural quantum analogues of the classical symmetric powers of a module over a complex semisimple Lie algebra. We show that their point modules form natural non-commutative curves and surfaces and conjecture that braided symmetric algebras give rise to an interesting non-commutative geometry, which can be viewed as a flat deformation of the geometry associated to their classical limits.