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Combinatorics of infinite rank module categories over finite dimensional $\mathfrak{sl}_3$-modules in Lie-algebraic context

Volodymyr Mazorchuk, Xiaoyu Zhu

TL;DR

This paper addresses the problem of classifying transitive module categories over the monoidal category $\mathscr{C}$ of finite dimensional $\mathfrak{sl}_3$-modules acting on arbitrary simple $\mathfrak{sl}_3$-modules. It develops a Lie-theoretic framework using Harish-Chandra bimodules to reduce the classification to two natural sources of simple modules: category $\mathcal{O}$ and Whittaker modules, yielding eight infinite graphs that govern the combinatorics of all simple subquotients. The authors compute Perron–Frobenius eigenvectors with eigenvalue $3$ for these graphs and interpret the coefficients via dimensions, Gelfand–Kirillov dimensions, and Bernstein numbers, tying the combinatorics to Lie-theoretic data. The results unify the $\mathfrak{sl}_3$ case with the prior $\mathfrak{sl}_2$ work, and provide a comprehensive Lie-algebraic classification of transitive $\mathscr{C}$-module subquotients, including explicit treatments of category $\mathcal{O}$ blocks, partially integral weights, and Whittaker modules, with implications for deeper structural aspects of module categories and their Grothendieck groups.

Abstract

We determine the combinatorics of transitive module categories over the monoidal category of finite dimensional $\mathfrak{sl}_3$-modules which arise when acting by the latter monoidal category on arbitrary simple $\mathfrak{sl}_3$-modules. This gives us a family of eight graphs which can be viewed as $\mathfrak{sl}_3$-generalizations of the classical infinite Dynkin diagrams.

Combinatorics of infinite rank module categories over finite dimensional $\mathfrak{sl}_3$-modules in Lie-algebraic context

TL;DR

This paper addresses the problem of classifying transitive module categories over the monoidal category of finite dimensional -modules acting on arbitrary simple -modules. It develops a Lie-theoretic framework using Harish-Chandra bimodules to reduce the classification to two natural sources of simple modules: category and Whittaker modules, yielding eight infinite graphs that govern the combinatorics of all simple subquotients. The authors compute Perron–Frobenius eigenvectors with eigenvalue for these graphs and interpret the coefficients via dimensions, Gelfand–Kirillov dimensions, and Bernstein numbers, tying the combinatorics to Lie-theoretic data. The results unify the case with the prior work, and provide a comprehensive Lie-algebraic classification of transitive -module subquotients, including explicit treatments of category blocks, partially integral weights, and Whittaker modules, with implications for deeper structural aspects of module categories and their Grothendieck groups.

Abstract

We determine the combinatorics of transitive module categories over the monoidal category of finite dimensional -modules which arise when acting by the latter monoidal category on arbitrary simple -modules. This gives us a family of eight graphs which can be viewed as -generalizations of the classical infinite Dynkin diagrams.
Paper Structure (43 sections, 20 theorems, 39 equations, 17 figures)

This paper contains 43 sections, 20 theorems, 39 equations, 17 figures.

Table of Contents

  1. Introduction and description of the results
  2. Infinite Dynkin diagrams
  3. $\mathfrak{sl}_3$-generalization
  4. Content of the paper
  5. Structure of the paper
  6. The Lie algebra $\mathfrak{sl}_3$
  7. Setup
  8. Weights and roots
  9. Highest weight modules
  10. Finite dimensional $\mathfrak{sl}_3$-modules
  11. Weights of finite dimensional $\mathfrak{sl}_3$-modules
  12. Gelfand-Kirillov dimension and Bernstein number
  13. Projective functors
  14. Locally finitary $\mathscr{C}$-module categories
  15. Basics
  16. ...and 28 more sections

Key Result

Theorem 1

Let $\mathcal{M}$ be an admissible simple $\mathscr{C}$-module category with the same graph and dual graph as ${}_\mathscr{C}\mathscr{C}$. Then $\mathcal{M}$ is equivalent to ${}_\mathscr{C}\mathscr{C}$.

Figures (17)

  • Figure 1: The sets $\Lambda$, $\Theta$ and $\mathbf{R}$
  • Figure 2: The supports of $L((1,0))$, $L((0,1))$ and $L((1,1))$
  • Figure 3: The graph of the left regular $\mathscr{C}$-module category and the corresponding eigenvector
  • Figure 4: Top, middle and bottom integral weights
  • Figure 5: The graph of the upper middle $\mathscr{C}$-module category $\mathcal{N}_1$ and the corresponding positive eigenvector
  • ...and 12 more figures

Theorems & Definitions (40)

  • Theorem 1
  • proof
  • Remark 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • ...and 30 more