Spectral analysis of finite dimensional algebras and singularities

By H. Lenzing, José Antonio de la Peña. 2008

65 citations recorded by the source corpus.

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 �A. 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 categorifica- tion of the Milnor lattice through triangulated categories which are naturally attached to a weighted projective line.

Subfields: Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/40e1a8c1e0f591a52f84564d09a486658a1e414b

Citations in the corpus

• Triangle singularities, ADE-chains, and weighted projective lines (2012, background): “This allows the following attempt in to ’define’ the notion of an ADE-chain (Tn) of triangulated categories Tn, by requesting the three properties below: (1) The sequence (Tn) forms an infinite chain T1 ⊂ T2 ⊂ T2 ⊂ · · · of triangulated categories with Serre duality which are fractionally Calabi-Yau”