Unitarizable representations of quivers

TL;DR

This paper studies unitarizable representations of quivers and associated *-algebras, providing classifications and parameter estimates for irreducible unitary representations, with implications for stability and construction methods.

Contribution

It offers an ADE-classification for unitarization of representations of unbound quivers and methods to construct unitarizable representations from polystable ones in bound quivers.

Findings

ADE-classification of unitarizable representations for unbound quivers

Construction methods for unitarizable representations in bound quivers

Estimate of parameters for irreducible unitary representations

Abstract

We investigate representations of *-algebras associated with posets. Unitarizable representations of the corresponding (bound) quivers (which are polystable representations for some appropriately chosen slope function) give rise to representations of these algebras. Considering posets which correspond to unbound quivers this leads to an ADE-classification which describes the unitarization behaviour of their representations. Considering posets which correspond to bound quivers, it is possible to construct unitarizable representations starting with polystable representations of related unbound quivers which can be glued together with a suitable direct sum of simple representations. Finally, we estimate the number of complex parameters parametrizing irreducible unitary non-equivalent representations of the corresponding algebras.