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Stable maps, Q-operators and category O

David Hernandez

Abstract

Motivated by Maulik-Okounkov stable maps associated to quiver varieties, we define and construct algebraic stable maps on tensor products of representations in the category O of the Borel subalgebra of an arbitrary untwisted quantum affine algebra. Our representation-theoretical construction is based on the study of the action of Cartan-Drinfeld subalgebras. We prove the algebraic stable maps are invertible and depend rationally on the spectral parameter. As an application, we obtain new R-matrices in the category O and we establish that a large family of simple modules, including the prefundamental representations associated to Q-operators, generically commute as representations of the Cartan-Drinfeld subalgebra. We also establish categorified QQ*-systems in terms of the R-matrices we construct.

Stable maps, Q-operators and category O

Abstract

Motivated by Maulik-Okounkov stable maps associated to quiver varieties, we define and construct algebraic stable maps on tensor products of representations in the category O of the Borel subalgebra of an arbitrary untwisted quantum affine algebra. Our representation-theoretical construction is based on the study of the action of Cartan-Drinfeld subalgebras. We prove the algebraic stable maps are invertible and depend rationally on the spectral parameter. As an application, we obtain new R-matrices in the category O and we establish that a large family of simple modules, including the prefundamental representations associated to Q-operators, generically commute as representations of the Cartan-Drinfeld subalgebra. We also establish categorified QQ*-systems in terms of the R-matrices we construct.

Paper Structure

This paper contains 21 sections, 19 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Background on quantum affine algebras
  3. Quantum affine algebras and Borel algebras
  4. Category $\mathcal{O}$ for representations of Borel algebras
  5. Finite-dimensional representations
  6. $\ell$-weight spaces
  7. Algebraic stable maps
  8. Definition and construction
  9. Deformations
  10. Examples
  11. Normalized stable map
  12. $R$-matrices and finite-dimensional representations
  13. Reminders on $R$-matrices
  14. Relations to known constructions
  15. $R$-matrices in the category $\mathcal{O}$
  16. ...and 6 more sections

Key Result

Proposition 2.5

For any $\hbox{\boldmath$\Psi$}\in \mathbf{\mathfrak{t}}^\times_\ell$, there exists a simple highest $\ell$-weight module $L(\hbox{\boldmath$\Psi$})$ of highest $\ell$-weight $\hbox{\boldmath$\Psi$}$. This module is unique up to isomorphism.

Theorems & Definitions (70)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9
  • Example 2.10
  • ...and 60 more