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

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.

24 pages

References in corpus (2)

• The Integral quantum loop algebra of ${\mathfrak{gl}}_n$
• Multiplicaton formulas and canonical basis for quantum affine gl_n