Remarks on PBW bases of Ringel-Hall algebras of cyclic quivers

TL;DR

This paper develops recursive formulas and new constructions for PBW and canonical bases of Ringel-Hall algebras of cyclic quivers, and explores their relations to affine quantum Schur algebras.

Contribution

It introduces recursive formulas for PBW bases, constructs canonical bases from monomial bases, and analyzes their relations to affine quantum Schur algebras.

Findings

Recursive formula for PBW bases of cyclic quiver Hall algebras.

Construction of canonical bases from monomial bases.

Complete determination of canonical bases for ^+(\u00f8)_2 with modules of Lowery length b3.

Abstract

In this paper, we give a recursive formula for the interesting PBW basis $E_{A}$ of composition subalgebras of Ringel-Hall algebras $\fkH_\vartri(n)$ of cyclic quivers after \cite{DengDuXiao2007generic}, and another construction of canonical bases of $\U_v^+(\wih{\fks\fkl}_n)$ from the monomial bases $m^{(A)}$ follow \cite{DuZhaomultiplication}. As an application, we will determined all the canonical basis of $\U_v^+(\wih{\fks\fkl}_2)$ associated with modules of Lowery length $\leqs3$. Finally, we will discuss the relation of canonical bases of Ringel-Hall algebras and those of affine quantum Schur algebras.