On derived Hall numbers for tame quivers

TL;DR

This paper investigates the derived Hall algebra of the bounded derived category of nilpotent representations of tame quivers over finite fields, showing that derived Hall numbers are rational functions of the field size.

Contribution

It establishes that derived Hall numbers for tame quivers are rational functions in the size of the finite field, extending understanding of their algebraic structure.

Findings

Derived Hall numbers are rational functions of the ground field size.

The study applies to the bounded derived category of nilpotent representations.

Provides a formula for derived Hall numbers in the context of tame quivers.

Abstract

In the present paper we study the derived Hall algebra for the bounded derived category of the nilpotent representations of a tame quiver over a finite field. We show that for any three given objects in the bounded derived category, the associated derived Hall numbers are given by a rational function in the cardinalities of ground fields.