Structure coefficients for quantum groups

TL;DR

This paper presents geometric realizations of structure coefficients for quantum groups using quiver moduli spaces, offering new proofs and generalizations within the framework of Hall algebras and Lusztig's construction.

Contribution

It introduces a geometric approach to structure coefficients, provides an alternative proof of Hall polynomials, and generalizes the Reineke-Caldero expression for the bar involution.

Findings

Geometric realization of coefficients via standard sheaves on quiver moduli spaces.

Alternative proof of the existence of Hall polynomials.

Generalization of the Reineke-Caldero expression for symmetrizable cases.

Abstract

According to the Hall algebras of quivers with automorphisms under Lusztig's construction, the polynominal forms of several structure coefficients for quantum groups of all finite types are presented in this note. We first provide a geometric realization of the coefficients between PBW basis and the canonical basis via standard sheaves on quiver moduli spaces with admissible automorphisms. This realization is constructed through Lusztig sheaves equipped with periodic functors and their modified Grothendieck groups. Second, within this geometric framework, we present an alternative proof for the existence of Hall polynomials originally due to Ringel. Finally, we give a slight generalization of the Reineke-Caldero expression for the bar involution of PBW basis elements in symmetrizable cases. When the periodic functor $\mathbf{a}^*$ is taken $\operatorname{id}$, our results are the same as Lusztig's and Caldero-Reineke's.