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Non-commutative cluster varieties

Algebraic Geometry

Speaker: Alexander Goncharov, Yale University/MSRI
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Location: 2112 MSB
Start time: Wed, Feb 8 2023, 4:10PM

Let R be a non-commutative field. We show that generic triples of flags in an m-dimensional R-vector space are described by flat R-line bundles on the honeycomb graph with (m-1)(m-2)/2 holes.

A similar description of generic n-tuples of flags leads to a non-commutative analog of cluster Poisson varieties. Instead of honeycomb graphs we use certain bipartite graphs; flat R-line bundles on any of them serve as a non-commutative cluster Poisson torus; special moves relating the graphs give rise to non-commutative cluster Poisson transformations.

In a parallel story for n decorated flags we get non-commutative cluster A-coordinates, calculated via Gelfand-Retakh quasideterminants, providing a geometric framework for them.

The talk is based on our joint work with Maxim Kontsevich.