Abstract representation theory of quivers and spectral Picard groups

Abstract: While the (derived) representation theory of quivers over a field is by now well-understood, much less is known when moving to coefficients in the integers or an arbitrary commutative ring. In this talk, we take a rather radical but well-founded approach: it has recently been observed that certain well-known symmetries of categories of representations (tilting results) are actually mere consequences of the stability of the coefficients involved, and so they exist in a much broader generality, often for the corresponding representations in any stable homotopy theory — this includes arbitrary rings, schemes, dg algebras, or ring spectra. For a finite acyclic quiver Q, we present here a method for producing universal autoequivalences of representations C^Q in any stable ∞-category C, which are the elements of the spectral Picard group of Q. This is based on an abstract equivalence of C^Q with a certain mesh ∞-category of representations of the Auslander–Reiten quiver Γ_Q. Then our universal equivalences arise from symmetries of Γ_Q, and thus yield abstract versions of key functors in classical representation theory — e.g. the Auslander-Reiten translation, the Serre functor, etc. Moreover, for representations of trees this allows us to realize the whole derived Picard group over a field as a factor of the spectral Picard group.

mathematical physicscommutative algebraalgebraic geometryrings and algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic

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Longitudinal Algebra and Geometry Open ONline Seminar (LAGOON)

Series comments: Description: Research webinar series

The LAGOON webinar series was supported as part of the International Centre for Mathematical Sciences (ICMS) and the Isaac Newton Institute for Mathematical Sciences (INI) Online Mathematical Sciences Seminars and is now sponsored by the University of Cologne and the University of Pavia. Please register here to obtain the Zoom link and password for the meetings.

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