Combinatorial classification of piecewise hereditary algebras

TL;DR

This paper advances the classification of piecewise hereditary algebras by utilizing the characteristic polynomial of their Coxeter matrices and offers a cohomological perspective on its coefficients.

Contribution

It introduces a new combinatorial classification method based on the characteristic polynomial and provides a cohomological interpretation of its coefficients.

Findings

Complete classification of piecewise hereditary algebras using characteristic polynomials

Cohomological interpretation of Coxeter matrix polynomial coefficients

Extension of Happel's classification framework

Abstract

We use the characteristic polynomial of the Coxeter matrix of an algebra to complete the combinatorial classification of piecewise hereditary algebras which Happel gave in terms of the trace of the Coxeter matrix. We also give a cohomological interpretation of the coefficients (other than the trace) of the characteristic polynomial of the Coxeter matrix of any finite dimensional algebra with finite global dimension.