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Semisimple module categories with fusion rules of the compact full flag manifold type

Mao Hoshino

TL;DR

This work classifies semisimple left module categories over the fusion category $\mathrm{Rep}_q^{\mathrm{f}} G$ with fusion rules arising from the maximal torus, linking operator-algebraic equivariant quantizations to equivariant Poisson structures on compact full flag manifolds and coadjoint orbits. The authors develop an explicit construction using deformed quantum enveloping algebras via the category $\mathcal{O}_{q,\chi}$, and establish precise unitarizability conditions. They prove algebraic and operator-algebraic classification theorems for actions of $H\backslash SL_n$-type and $T\backslash SU(n)$-type, expressed through unique parameters in root-system moduli spaces, and show how invariant coefficients distinguish inequivalent actions. The non-quantum case is analyzed to reveal no nontrivial equivariant quantization in the operator-algebraic setting, highlighting a no-go phenomenon aligned with Poisson-geometry constraints. Overall, the paper unifies Poisson geometry, quantum groups, and tensor-categorical methods to classify semisimple actions of flag-manifold-type on both algebraic and operator-algebraic levels.

Abstract

We classify semisimple left module categories over the representation category of a type A quantum group whose fusion rules arise from the maximal torus. The classification is connected to equivariant Poisson structures on compact full flag manifolds in the operator-algebraic setting, and on semisimple coadjoint orbits in the algebraic setting. We also provide an explicit construction based on the BGG categories of deformed quantum enveloping algebras, whose unitarizability corresponds to being of quotient type. Finally, we present a brief discussion of the non-quantum case.

Semisimple module categories with fusion rules of the compact full flag manifold type

TL;DR

This work classifies semisimple left module categories over the fusion category with fusion rules arising from the maximal torus, linking operator-algebraic equivariant quantizations to equivariant Poisson structures on compact full flag manifolds and coadjoint orbits. The authors develop an explicit construction using deformed quantum enveloping algebras via the category , and establish precise unitarizability conditions. They prove algebraic and operator-algebraic classification theorems for actions of -type and -type, expressed through unique parameters in root-system moduli spaces, and show how invariant coefficients distinguish inequivalent actions. The non-quantum case is analyzed to reveal no nontrivial equivariant quantization in the operator-algebraic setting, highlighting a no-go phenomenon aligned with Poisson-geometry constraints. Overall, the paper unifies Poisson geometry, quantum groups, and tensor-categorical methods to classify semisimple actions of flag-manifold-type on both algebraic and operator-algebraic levels.

Abstract

We classify semisimple left module categories over the representation category of a type A quantum group whose fusion rules arise from the maximal torus. The classification is connected to equivariant Poisson structures on compact full flag manifolds in the operator-algebraic setting, and on semisimple coadjoint orbits in the algebraic setting. We also provide an explicit construction based on the BGG categories of deformed quantum enveloping algebras, whose unitarizability corresponds to being of quotient type. Finally, we present a brief discussion of the non-quantum case.
Paper Structure (24 sections, 62 theorems, 191 equations)

This paper contains 24 sections, 62 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and convensions
  4. Lie algebras and Lie groups
  5. The Drinfeld-Jimbo deformations
  6. Compact real forms
  7. Equivariant Poisson structures on compact flag manifolds
  8. A brief review on the classification theorem for $L_S \backslash G$
  9. The classification theorem for $K_S\backslash K$
  10. The toric variety associated to a root system
  11. The category $\mathcal{O}$ for deformed QEA
  12. Deformed quantum enveloping algebras
  13. The category $\mathcal{O}_{q,\chi}$
  14. Highest weight vectors in tensor products
  15. Actions of $H\backslash G$-type and $T\backslash K$-type
  16. ...and 9 more sections

Key Result

Proposition 3.1

Let $\pi$ be a bivector field on $L_S\backslash G$. Then $(L_S\backslash G, \pi)$ is a Poisson $G^{\mathrm{std}}$-variety if and only if $\pi = v(\varphi)_R - r_L$ for some $\varphi \in X_{L_S\backslash G}(k)$.

Theorems & Definitions (133)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof : Proof of Proposition \ref{['prop:characterization of quotient Poisson structure']} (i) $\Longleftrightarrow$ (ii) $\Longrightarrow$ (iii)
  • Lemma 3.4
  • Lemma 3.5
  • proof
  • proof : Proof of Proposition \ref{['prop:characterization of quotient Poisson structure']} (ii) $\Longrightarrow$ (i)
  • Lemma 3.6
  • Lemma 4.1: c.f. MR4837934
  • ...and 123 more