Elementary construction of canonical bases, foldings, and piecewise linear bijections

TL;DR

The paper presents an elementary construction of the canonical basis for the negative half $\mathbf U_q^-$ of a finite-type quantum group by leveraging folding theory and piecewise linear parametrizations of PBW bases. It replaces reliance on Lusztig's geometric theory or Kashiwara's crystal bases with a direct comparison to the folded algebra $\ul{\mathbf U}_q^-$ via an admissible automorphism, using a folding isomorphism $\Phi$ and projections to transfer bases. A central contribution is showing that, in type $B_2$, the $\sigma$-fixed canonical labeling is independent of the chosen reduced expression, established through explicit piecewise-linear bijections (L-piece) and detailed combinatorial arguments. This yields an self-contained, algebraic route to canonical bases in finite type, with explicit maps connecting different PBW parametrizations and a reduction to solvable rank-2 cases. The approach broadens the toolkit for canonical bases by providing concrete, elementary constructions and folding-based comparisons that avoid deep geometric or crystal-theoretic machinery.

Abstract

Let ${\mathbf U}_q^-$ be the negative half of a quantum group of finite type. We construct the canonical basis of ${\mathbf U}_q^-$ by applying the folding theory of quantum groups, and piecewise linear parametrization of canonical basis. Our construction is elementary, in the sense that we don't appeal to Lusztig's geometric theory of canonical bases, nor to Kashiwara's theory of crystal bases.

Theorems & Definitions (19)

• Proposition 1.4
• Theorem 2.3: [SZ1]
• Proposition 2.7
• Theorem 2.10
• Proposition 3.5
• Lemma 3.6
• Lemma 3.7
• proof
• Proposition 3.9
• Lemma 3.11