Non-commutative Poisson geometry of Boalch's multiplicative quiver varieties

Maxime Fairon (University of Glasgow)

Wednesday 19th January, 2022 16:00-17:00 Maths 110


Multiplicative quiver varieties form an interesting class of moduli spaces constructed by Crawley-Boevey and Shaw using quivers (i.e. directed graphs). Shortly after their introduction, it was observed by Van den Bergh that these spaces are equipped with a Poisson structure, which is obtained by reduction from a quasi-Poisson bracket. Interestingly, Van den Bergh noticed that this process could be understood directly at the level of the quivers, using a non-commutative version of quasi-Poisson geometry. My first aim is to review this construction, starting with the non-commutative Poisson geometry of quiver varieties. My second aims is to state a conjecture generalising Van den Bergh's result. In that case, we are interested in a class of varieties introduced by Boalch which encompasses multiplicative quiver varieties, as well as some wild character varieties. The goal is to understand the Poisson geometry of these spaces directly at the level of quivers endowed with a colour function. This is based on joint work with David Fernández (arXiv:2103.10117).

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