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PBW theory for Bosonic extensions of quantum groups

Se-jin Oh, Euiyong Park

Abstract

In this paper, we develop the PBW theory for the bosonic extension $\qbA{\g}$ of a quantum group $\mathcal{U}_q(\g)$ of \emph{any} finite type. When $\g$ belongs to the class of \emph{simply-laced type}, the algebra $\qbA{\g}$ arises from the quantum Grothendieck ring of the Hernandez-Leclerc category over quantum affine algebras of untwisted affine types. We introduce and investigate a symmetric bilinear form $\pair{\ , \ }$ on $\qbA{\g}$ which is invariant under the braid group actions $\bT_i$ on $\qbA{\g}$, and study the adjoint operators $\Ep_{i,p}$ and $\Es_{i,p}$ with respect to $\pair{\ , \ }$. It turns out that the adjoint operators $\Ep_{i,p}$ and $\Es_{i,p}$ are analogues of the $q$-derivations $e_i'$ and $\es_i$ on the negative half $\calU_q^-(\g)$ of $\calU_q(\g)$. Following this, we introduce a new family of subalgebras denoted as $\qbA{\mathfrak{g}}(\ttb)$ in $\qbA{\mathfrak{g}}$. These subalgebras are defined for any elements $\ttb$ in the positive submonoid $\bg^+$ of the (generalized) braid group $\ttB$ of $\g$. We prove that $\qbA{\mathfrak{g}}(\ttb)$ exhibits PBW root vectors and PBW bases defined by $\bT_\ii$ for any sequence $\ii$ of $\ttb$. The PBW root vectors satisfy a Levendorskii-Soibelman formula and the PBW bases are orthogonal with respect to $\pair{\ , \ }$. The algebras $\qbA{\g} (\ttb)$ can be understood as a natural extension of quantum unipotent coordinate rings.

PBW theory for Bosonic extensions of quantum groups

Abstract

In this paper, we develop the PBW theory for the bosonic extension of a quantum group of \emph{any} finite type. When belongs to the class of \emph{simply-laced type}, the algebra arises from the quantum Grothendieck ring of the Hernandez-Leclerc category over quantum affine algebras of untwisted affine types. We introduce and investigate a symmetric bilinear form on which is invariant under the braid group actions on , and study the adjoint operators and with respect to . It turns out that the adjoint operators and are analogues of the -derivations and on the negative half of . Following this, we introduce a new family of subalgebras denoted as in . These subalgebras are defined for any elements in the positive submonoid of the (generalized) braid group of . We prove that exhibits PBW root vectors and PBW bases defined by for any sequence of . The PBW root vectors satisfy a Levendorskii-Soibelman formula and the PBW bases are orthogonal with respect to . The algebras can be understood as a natural extension of quantum unipotent coordinate rings.
Paper Structure (18 sections, 33 theorems, 138 equations)

This paper contains 18 sections, 33 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum groups
  4. Braid groups and Weyl groups
  5. PBW bases for $\mathcal{U}_q^-(\mathfrak{g})$
  6. Bilinear forms and $q$-boson algebras
  7. Bosonic extensions of quantum groups
  8. Braid group actions and bilinear forms
  9. Braid group actions
  10. Bilinear form on ${\widehat{\mathcal{A}}_{}}$
  11. PBW bases: locally reduced sequences
  12. Case for ${\widehat{\mathcal{A}}_{[a,b]}}$
  13. Case for ${\widehat{\mathcal{A}}_{}}$
  14. Adjoint operators $E'_{i,k}$ and $E^{\star}_{i,k}$
  15. PBW bases: arbitrary sequences
  16. ...and 3 more sections

Key Result

Lemma 2.3

For elements $y,z \in \mathcal{U}^-_q(\mathfrak{g})$ with $e^\star_i(y)=0$ and $e'_i(z)=0$, we have for any $m \in \mathbb{Z}\mspace{1mu}_{\geqslant 0}$.

Theorems & Definitions (72)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 3.1: see HL15, JLO2
  • Remark 3.2
  • Lemma 3.3
  • proof
  • ...and 62 more