Some Identities in Quantum Torus Arising from Ringel-Hall Algebras

TL;DR

This paper establishes new identities in the quantum torus by analyzing monomorphic and epimorphic representations of quivers, revealing structural decompositions and their algebraic implications.

Contribution

It introduces a novel approach to derive identities in the quantum torus from the structure of quiver representations and their maximal subrepresentations.

Findings

Unique maximal nilpotent subrepresentations exist for all representations.

Every representation has a unique maximal monomorphic subrepresentation.

Two algebraic identities in the Ringel-Hall algebra and quantum torus are derived.

Abstract

We define two classes of representations of quivers over arbitrary fields, called monomorphic representations and epimorphic representations. We show that every representation has a unique maximal nilpotent subrepresentation and the associated quotient is always monomorphic, and every representation has a unique maximal epimorphic subrepresentation and the associated quotient is always nilpotent. The uniquenesses of such subrepresenations imply two identities in the Ringel-Hall algebra. By applying Reineke's integration map, we obtain two identities in the corresponding quantum torus.