Non-perturbative symplectic manifolds and non-commutative algebras


Philip Boalch

10:00:00 - 10:50:00

101 , Mathematics Research Center Building (ori. New Math. Bldg.)

From a moduli-theoretic viewpoint the irregular Riemann-Hilbert correspondence can be viewed as a machine that takes as input a simple “additive” symplectic/Poisson manifold and outputs a more complicated “multiplicative” symplectic/Poisson manifold (a wild character variety). Since 2000 (arXiv:math/0011062) it has been understood that this machine returns the dual Poisson Lie group G* (the nonlinear Poisson manifold underlying the Drinfeld-Jimbo quantum group) when the input is the linear Poisson manifold Lie(G)*. In this talk I will describe some more recent (and more complicated) examples.

For example in 2008 (arXiv:0806.1050) it was understood that a large class of Nakajima quiver varieties (often far from the affine case) can be taken as input. If we look at their non-perturbative versions (arXiv:1307.1033) we see a generalisation of the theory of multiplicative quiver varieties. In turn this yields some new noncommutative algebras “fission algebras” which generalise the deformed mulitplicative preprojective algebras of Crawley-Boevey and Shaw (simple cases of which contain the generalised double affine Hecke algebras of Etingof-Oblomkov-Rains).

We obtain new noncommutative algebras even in the case of the triangle and the doubled edge (affine A2 and affine A1). Much of this work involves the wild extension (started in 2002 arXiv:0203161) of the quasi-Hamiltonian geometry of Alekseev-Malkin-Meinrenken.