Spectral analysis of finite dimensional algebras and singularities

  • Luz de Teresa

    Universidad Nacional Autonoma de México
  • Helmut Lenzing

    Universität Paderborn, Germany
Spectral analysis of finite dimensional algebras and singularities cover

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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.