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On cacti and crystals

Arkady Berenstein, Jacob Greenstein, Jian-Rong Li

Abstract

In the present work we study actions of various groups generated by involutions on the category $\mathscr O^{int}_q(\mathfrak g)$ of integrable highest weight $U_q(\mathfrak g)$-modules and their crystal bases for any symmetrizable Kac-Moody algebra $\mathfrak g$. The most notable of them are the cactus group and (yet conjectural) Weyl group action on any highest weight integrable module and its lower and upper crystal bases. Surprisingly, some generators of cactus groups are anti-involutions of the Gelfand-Kirillov model for $\mathscr O^{int}_q(\mathfrak g)$ closely related to the remarkable quantum twists discovered by Kimura and Oya.

On cacti and crystals

Abstract

In the present work we study actions of various groups generated by involutions on the category of integrable highest weight -modules and their crystal bases for any symmetrizable Kac-Moody algebra . The most notable of them are the cactus group and (yet conjectural) Weyl group action on any highest weight integrable module and its lower and upper crystal bases. Surprisingly, some generators of cactus groups are anti-involutions of the Gelfand-Kirillov model for closely related to the remarkable quantum twists discovered by Kimura and Oya.

Paper Structure

This paper contains 27 sections, 105 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coxeter groups
  4. Cartan data and Weyl group
  5. Quantum groups
  6. Integrable modules
  7. Crystal operators, lattices and bases
  8. Properties of eta i and proof of Theorem \ref{['thm:crystal weyl group']}
  9. Special monomials in U_q +-(g)
  10. Extremal vectors
  11. Proof of Theorem \ref{['thm:crystal weyl group']}
  12. Modified Lusztig symmetries and involutions eta J
  13. Modified Lusztig's symmetries
  14. eta via modified Lusztig's symmetries
  15. Parabolic involutions and proof of Theorem \ref{['thm:main thm']}
  16. ...and 12 more sections

Key Result

Theorem \oldthetheorem

For any non-zero module $V\in\mathscr O^{int}_q({\mathfrak{g}})$, the assignments where $J(V)=\{ i\in I\,:\, F_i(V)=\{0\}\}$, define a homomorphism $\psi_V$ from $\mathsf W(V)$ to the Weyl group $W$ of ${\mathfrak{g}}$.

Theorems & Definitions (177)

  • Theorem \oldthetheorem
  • Conjecture \oldthetheorem
  • Remark \oldthetheorem
  • Proposition \oldthetheorem: Proposition \ref{['prop:prop eta^J']}
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem: Theorem \ref{['thm:twist']}
  • ...and 167 more