Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$\mathfrak b_1$-Verma $\mathfrak b_2$-dual Verma supermodules

Shunsuke Hirota

TL;DR

The paper analyzes when a module $M$ that is simultaneously a $\mathfrak b_1$-Verma module and a $\mathfrak b_2$-dual Verma module must itself be a Verma module for a distinguished Borel in a basic classical Lie superalgebra of type I. It introduces edge contractions on the finite Young lattice $L(m,n)$ to model odd reflections, and proves that such an $M$ is isomorphic to a Verma module for either the distinguished or the anti-distinguished Borel, leveraging a precise description of bimodule homomorphisms along rainbow shortest walks (Odd Verma's theorem). The results hinge on the combinatorics of the odd reflection graph and establish a bridge between dual Verma structures and standard Verma realizations, with the method extensible to other basic classical Lie superalgebras via the Weyl groupoid framework. The findings illuminate how changes of Borel subalgebras interact with Verma flags in category $\mathcal{O}$, and open a combinatorial avenue for understanding antidominance and dualities in super representation theory.

Abstract

We show that if a module M over a basic classical Lie superalgebra of type type I is simultaneously a Verma module with respect to some Borel \(\mathfrak b_1\) and a dual Verma module with respect to Borel \(\mathfrak b_2\), then M is isomorphic to a Verma module with respect to either distinguished or an anti-distinguished Borel. Our method proceeds by analyzing edge contractions of the finite Young lattice that controls the combinatorics of odd reflections. In principle, the same strategy, for the most part, applies to all basic classical Lie superalgebras.

$\mathfrak b_1$-Verma $\mathfrak b_2$-dual Verma supermodules

TL;DR

The paper analyzes when a module that is simultaneously a -Verma module and a -dual Verma module must itself be a Verma module for a distinguished Borel in a basic classical Lie superalgebra of type I. It introduces edge contractions on the finite Young lattice to model odd reflections, and proves that such an is isomorphic to a Verma module for either the distinguished or the anti-distinguished Borel, leveraging a precise description of bimodule homomorphisms along rainbow shortest walks (Odd Verma's theorem). The results hinge on the combinatorics of the odd reflection graph and establish a bridge between dual Verma structures and standard Verma realizations, with the method extensible to other basic classical Lie superalgebras via the Weyl groupoid framework. The findings illuminate how changes of Borel subalgebras interact with Verma flags in category , and open a combinatorial avenue for understanding antidominance and dualities in super representation theory.

Abstract

We show that if a module M over a basic classical Lie superalgebra of type type I is simultaneously a Verma module with respect to some Borel and a dual Verma module with respect to Borel , then M is isomorphic to a Verma module with respect to either distinguished or an anti-distinguished Borel. Our method proceeds by analyzing edge contractions of the finite Young lattice that controls the combinatorics of odd reflections. In principle, the same strategy, for the most part, applies to all basic classical Lie superalgebras.

Paper Structure

This paper contains 17 sections, 26 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Well known basics
  4. Root systems
  5. Finite Young lattices $L(m,n)$
  6. Contragredient duality
  7. Verma modules
  8. Category $\mathcal{O}$
  9. Kac functors
  10. Odd reflection graph of Verma modules
  11. Edge contraction
  12. Odd Verma's theorem
  13. Main results
  14. Applying odd Verma's theorem
  15. A bridge in odd reflection graphs
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\mathfrak g=\mathfrak{gl}(m|n)$ and $\lambda$ be an integral weight and let $\mathfrak b_1$ be a Borel subalgebra. If $M^{\mathfrak b_1}(\lambda)\cong M^{\mathfrak b_2}(\mu)^{\vee}$ for some $\mu$ and some Borel $\mathfrak b_2$ , then $M^{\mathfrak b_1}(\lambda)$ is isomorphic to a Verma module

Theorems & Definitions (48)

  • Theorem 1.1
  • Example 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5: Edge-colored finite Young's Lattice
  • Proposition 1
  • Corollary 1
  • Example 2
  • ...and 38 more