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iCanonical basis arising from quasi-split rank one iquantum group

Ziming Chen

TL;DR

The paper achieves an explicit construction of the icanonical basis for the quasi-split rank-one iquantum group by deriving detailed transition matrices between the icanonical, monomial, and standardized canonical bases on the modified algebra \dot{\bold{U}}^{\imath} and its simple modules. Central to the work are parity-sensitive case analyses that distinguish when coefficients live in \(\mathbb{Z}[q^{-1}]\) and how inverse transition matrices are formed, guided by rank-one reductions from \(U_q(\mathfrak{sl}_3)\) and existing module-level results. The authors establish stability and symmetry properties of the icanonical basis, connect the module-level expansions to the algebra-level framework via projective limits, and finally relate the icanonical basis to Lusztig's canonical basis through new explicit formulas. These results enable categorification and provide concrete, computable tools for understanding iquantum group representations in the quasi-split rank-one setting, with potential extension to higher-rank cases. Overall, the work fills a key gap in explicit basis transitions for quasi-split iquantum groups and strengthens the bridge between icanonical and canonical theories in the iquantum context.

Abstract

We compute icanonical basis of the quasi-split rank one modified iquantum group, by obtaining explicit transition matrices among the icanonical basis, monomial basis, and standardized canonical basis; all these bases can be naturally categorified. These transition matrices follow from their counterparts computed in this paper among the icanonical basis, monomial basis, and canonical basis on simple finite-dimensional modules of quantum $\mathfrak{sl}_3$.

iCanonical basis arising from quasi-split rank one iquantum group

TL;DR

The paper achieves an explicit construction of the icanonical basis for the quasi-split rank-one iquantum group by deriving detailed transition matrices between the icanonical, monomial, and standardized canonical bases on the modified algebra \dot{\bold{U}}^{\imath} and its simple modules. Central to the work are parity-sensitive case analyses that distinguish when coefficients live in and how inverse transition matrices are formed, guided by rank-one reductions from \(U_q(\mathfrak{sl}_3)\) and existing module-level results. The authors establish stability and symmetry properties of the icanonical basis, connect the module-level expansions to the algebra-level framework via projective limits, and finally relate the icanonical basis to Lusztig's canonical basis through new explicit formulas. These results enable categorification and provide concrete, computable tools for understanding iquantum group representations in the quasi-split rank-one setting, with potential extension to higher-rank cases. Overall, the work fills a key gap in explicit basis transitions for quasi-split iquantum groups and strengthens the bridge between icanonical and canonical theories in the iquantum context.

Abstract

We compute icanonical basis of the quasi-split rank one modified iquantum group, by obtaining explicit transition matrices among the icanonical basis, monomial basis, and standardized canonical basis; all these bases can be naturally categorified. These transition matrices follow from their counterparts computed in this paper among the icanonical basis, monomial basis, and canonical basis on simple finite-dimensional modules of quantum .
Paper Structure (44 sections, 66 theorems, 212 equations)

This paper contains 44 sections, 66 theorems, 212 equations.

Table of Contents

  1. Introduction
  2. Background
  3. What is achieved in this paper?
  4. Main results
  5. Organization
  6. Preliminaries
  7. Weight lattice and $\imath$weight lattice
  8. Quantum groups and iquantum groups
  9. Bilinear forms
  10. Canonical basis, icanonical basis, monomial basis, and standardized canonical basis
  11. Standardized canonical basis vs monomial basis
  12. iCanonical basis vs monomial basis: first observation
  13. The first observation
  14. Case for $a=0$, $(\varsigma_{1},\varsigma_{2})=(1,0)$, $\alpha_{b,c,\bm{\varsigma}}$ is even
  15. Case for $a=0$, $(\varsigma_{1},\varsigma_{2})=(1,0)$, $\alpha_{b,c,\bm{\varsigma}}$ is odd
  16. ...and 29 more sections

Key Result

Theorem A

Fix $\zeta\in X^{\imath}$, for any $a,b,c\geq 0$ such that $b\geq a+c$, we have the following identity on $\dot{\bold{U}}^{\imath}1_{\zeta}$: and

Theorems & Definitions (100)

  • Theorem A: Theorem \ref{['thm:stdCB1']}
  • Proposition B: Proposition \ref{['prop32']} and Theorem \ref{['thm711']}
  • Theorem C: Theorem \ref{['thm722']}
  • Theorem D: Theorem \ref{['thm733']}
  • Theorem 2.1: cf. WZ25
  • Theorem 2.2: cf. BWW25
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Lemma 4.1: cf. WZ25
  • ...and 90 more