Noncommutative geometry and Painlev\'e equations
Andrei Okounkov, Eric Rains
TL;DR
This paper constructs elliptic Painlevé equations and their higher-dimensional analogs using the framework of noncommutative geometry, specifically through the action of line bundles on sheaves on noncommutative surfaces.
Contribution
It introduces a novel geometric approach to formulating Painlevé equations within noncommutative geometry, expanding the understanding of their structure and symmetries.
Findings
Elliptic Painlevé equations are realized via line bundle actions on sheaves.
Higher-dimensional analogs of Painlevé equations are constructed.
The approach links noncommutative geometry with integrable systems.
Abstract
We construct the elliptic Painlev\'e equation and its higher dimensional analogs as the action of line bundles on 1-dimensional sheaves on noncommutative surfaces.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Algebraic Geometry and Number Theory · Advanced Topics in Algebra