Infinite-dimensional representations of $U_q(\mathfrak{sl}_2)$ and the shadow world: quantum $6j$-symbols for Verma modules

TL;DR

This paper explores invariants of links derived from infinite-dimensional representations of quantum groups, introducing new formulas for quantum 6j-symbols for Verma modules and examining their properties within the shadow world framework.

Contribution

It develops formulas for quantum 6j-symbols for Verma modules and investigates their properties, advancing the understanding of link invariants from infinite-dimensional quantum group representations.

Findings

Formulas for q3j- and q6j-symbols for Verma modules

Properties of these quantum 6j-symbols analyzed

Insights into the structure of the target category for invariants

Abstract

This paper initiates the study of invariants of links associated to infinite-dimensional representations of $U_q(\mathfrak{sl}_2)$ using graphical representation for quantum $6j$-symbols, the shadow world. We obtain formulae for $q3j$-symbols and $q6j$-symbols for Verma modules and study their properties. This hints at the structure of the possible target category for Reshetikhin-Turaev functor for such an invariant.