# Spectral analysis of finite dimensional algebras and singularities

### Luz de Teresa

Universidad Nacional Autonoma de México### Helmut Lenzing

Universität Paderborn, Germany

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## Abstract

For a finite dimensional algebra A of finite global dimension the bounded derived category of finite dimensional A-modules admits Auslander–Reiten triangles such that the Auslander–Reiten translation τ is an equivalence. On the level of the Grothendieck group τ induces the Coxeter transformation ΦA. More generally this extends to a homologically finite triangulated category **T** admitting Serre duality. In both cases the Coxeter polynomial, that is, the characteristic polynomial of the Coxeter transformation, yields an important homological invariant of A or **T**. Spectral analysis is the study of this interplay, it often reveals unexpected links between apparently different subjects.

This paper gives a summary on spectral techniques and studies the links to singularity theory. In particular, it offers a contribution to the categorification of the Milnor lattice through triangulated categories which are naturally attached to a weighted projective line.