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

Dmitry Solovyev

TL;DR

This work extends the shadow world framework from finite-dimensional $U_q(\mathfrak{sl}_2)$ representations to Verma modules by constructing explicit $q3j$-symbols and $q6j$-symbols. It develops a graphical calculus for these infinite-dimensional objects, proves key orthogonality, Racah, BE, and Yang–Baxter identities within the shadow world, and introduces the functional ${\mathsf{SW}}_\infty$ together with the notion of polarized modules as a potential target category for a Reshetikhin–Turaev-type theory. By connecting to the Gukov–Manolescu program via $F_K(x,q)$ and studying the trefoil as a diagnostic example, the paper lays out a roadmap for a non-perturbative, functorial framework that reconciles edge- and face-state perspectives in the infinite-dimensional setting. The results offer a pathway toward a categorified, topological invariant theory that encompasses Verma-module colorings and informs the structure of the Grothendieck ring for an RT-like functor in this broader representation-theoretic context.

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.

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

TL;DR

This work extends the shadow world framework from finite-dimensional representations to Verma modules by constructing explicit -symbols and -symbols. It develops a graphical calculus for these infinite-dimensional objects, proves key orthogonality, Racah, BE, and Yang–Baxter identities within the shadow world, and introduces the functional together with the notion of polarized modules as a potential target category for a Reshetikhin–Turaev-type theory. By connecting to the Gukov–Manolescu program via and studying the trefoil as a diagnostic example, the paper lays out a roadmap for a non-perturbative, functorial framework that reconciles edge- and face-state perspectives in the infinite-dimensional setting. The results offer a pathway toward a categorified, topological invariant theory that encompasses Verma-module colorings and informs the structure of the Grothendieck ring for an RT-like functor in this broader representation-theoretic context.

Abstract

This paper initiates the study of invariants of links associated to infinite-dimensional representations of using graphical representation for quantum -symbols, the shadow world. We obtain formulae for -symbols and -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.

Paper Structure

This paper contains 62 sections, 17 theorems, 182 equations, 29 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Results
  4. Acknowledgements
  5. Algebra $U_q(\mathfrak{sl}_2)$ and its representations
  6. Hopf algebra $U_q(\mathfrak{sl}_2)$
  7. The universal $R$-matrix
  8. Representations of $U_q(\mathfrak{sl}_2)$
  9. Representation maps $\pi_{M_\lambda}$ and $\pi_{M^*_\lambda}$
  10. Representation map $\pi_{M_\lambda}$
  11. Representation map $\pi_{M^*_\lambda}$
  12. Map $\phi:M_\lambda \to M^*_\lambda$
  13. The universal $R$-matrix in representation $\pi_{M_\lambda}$
  14. Quantum $3j$-symbols for Verma modules
  15. Definition of the $q3j$-symbols
  16. ...and 47 more sections

Key Result

Theorem 2.1

Module $M_\lambda$ is irreducible iff $\lambda$ is generic.

Figures (29)

  • Figure 1: Assignment of string diagrams for $\mathcal{R}$-matrix and $q3j$-symbols.
  • Figure 2: String diagrams for morphisms $A:M_{\underline{\lambda}^{\prime\prime}}\to M_{\underline{\lambda}}$ and $B:M_{\underline{\lambda}^{\prime}}\to M_{\underline{\lambda}^{\prime\prime}}$, where $A_{a_1,\ldots,a_m}^{a^{\prime\prime}_1,\ldots,a^{\prime\prime}_p}$ and $B^{a_1^\prime,\ldots,a_n^\prime}_{a^{\prime\prime}_1,\ldots,a^{\prime\prime}_p}$ are corresponding matrix elements.
  • Figure 3: String diagram for composition of morphisms $A$ and $B$.
  • Figure 4: String diagram representation for the Yang-Baxter relation (\ref{['YB']}).
  • Figure 5: String diagram representations for the orthagonality relation (\ref{['projth1']}).
  • ...and 24 more figures

Theorems & Definitions (39)

  • Theorem 2.1
  • Definition 2.2
  • Definition 3.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • proof
  • Theorem 4.4
  • ...and 29 more