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

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.

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