Nilpotent operators and weighted projective lines
By D. Kussin, H. Lenzing, H. Meltzer. 2010
78 citations recorded by the source corpus.
We show a surprising link between singularity theory and the invariant subspace problem of nilpotent operators as recently studied by C. M. Ringel and M. Schmidmeier, a problem with a longstanding history going back to G. Birkhoff. The link is established via weighted projective lines and (stable) categories of vector bundles on those. The setup yields a new approach to attack the subspace problem. In particular, we deduce the main results of Ringel and Schmidmeier for nilpotency degree p from properties of the category of vector bundles on the weighted projective line of weight type (2,3,p), obtained by Serre construction from the triangle singularity x^2+y^3+z^p. For p=6 the Ringel-Schmidmeier classification is thus covered by the classification of vector bundles for tubular type (2,3,6), and then is closely related to Atiyah's classification of vector bundles on a smooth elliptic curve. Returning to the general case, we establish that the stable categories associated to vector bundles or invariant subspaces of nilpotent operators may be naturally identified as triangulated categories. They satisfy Serre duality and also have tilting objects whose endomorphism rings play a role in singularity theory. In fact, we thus obtain a whole sequence of triangulated (fractional) Calabi-Yau categories, indexed by p, which naturally form an ADE-chain.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/690d45b9ef9a41187e60cd6255baddcde9225f14
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Auslander–Reiten translations in monomorphism categories (2011, background): “Meltzer [9] establish a surprising link between the stable submodule category with the singularity theory via weighted projective lines of type .”
- Monic representations and Gorenstein-projective modules (2011, background): “By [Kussin et al. 2012] it is also related to the singularity category; see also [Chen 2011].”
- Gorenstein Homological Aspects of Monomorphism Categories via Morita Rings (2015, background): “In fact, there are connections with classification problems (Ringel, Schmidmeier [40–42], Xiong, Zhang, Zhang [45]), with weighted projective lines (Kussin, Lenzing, Meltzer [31]), and with aspects of Gorenstein homological algebra (Beligiannis [10, 11], Zhang [49], Chen [17]).”
- Triangle singularities, ADE-chains, and weighted projective lines (2012, background): “In [32] we relate this problem to the stable category of vector bundles vect-X for weight type (2, 3, p), where p takes the role of the nilpotency degree.”