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Tame quivers and affine bases II: nonsimply-laced cases

Jie Xiao, Han Xu

Abstract

In [Tame_quivers_and_affine_bases_I], we give a Ringel-Hall algebra approach to the canonical bases in the symmetric affine cases. In this paper, we extend the results to general symmetrizable affine cases by using Ringel-Hall algebras of representations of a valued quiver. We obtain a bar-invariant basis $\mathbf{B}'=\{C(\mathbf{c},t_λ)|(\mathbf{c},t_λ)\in\mathcal{G}^a\}$ in the generic composition algebra $\mathcal{C}^*$ and prove that $\mathcal{B}'=\mathbf{B}'\sqcup(-\mathbf{B}')$ coincides with Lusztig's signed canonical basis $\mathcal{B}$. Moreover, in type $\tilde{B}_n,\tilde{C}_n$, $\mathbf{B}'$ is the canonical basis $\mathbf{B}$.

Tame quivers and affine bases II: nonsimply-laced cases

Abstract

In [Tame_quivers_and_affine_bases_I], we give a Ringel-Hall algebra approach to the canonical bases in the symmetric affine cases. In this paper, we extend the results to general symmetrizable affine cases by using Ringel-Hall algebras of representations of a valued quiver. We obtain a bar-invariant basis in the generic composition algebra and prove that coincides with Lusztig's signed canonical basis . Moreover, in type , is the canonical basis .
Paper Structure (48 sections, 49 theorems, 208 equations)

This paper contains 48 sections, 49 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Valued quivers
  4. Representations
  5. Cartan datum
  6. Root system and Weyl group
  7. Euler form
  8. Quantum groups
  9. Ringel-Hall algebras
  10. About $\hbox{\rm Rep}_k\Gamma$
  11. Affine and tame
  12. Frobenius folding
  13. Representations of a valued tame quiver
  14. Sequence $\underline{{\bf i}}$
  15. Preprojective and preinjective part
  16. ...and 33 more sections

Key Result

Theorem 1.1

For simply-laced affine type, we have $\Xi({\bf B}')={\bf B}"$. Moreover, we have $\Xi(C({\bf c},t_\lambda))=B({\bf c},t_\lambda)$.

Theorems & Definitions (69)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: Green_Hall_algebras_hereditary_algebras_and_quantum_groups
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3: Deng_Han
  • Definition 3.4
  • Proposition 4.1
  • proof
  • Corollary 4.2
  • ...and 59 more