On deformed double current algebras for simple Lie algebras

We prove the equivalence of two presentations of deformed double current algebras associated to a complex simple Lie algebra $\mathfrak{g}$, the first one obtained via a degeneration of affine Yangians while the other one naturally appeared in the construction of the elliptic Casimir connection. We also construct a specific central element of these algebras and, in type A, show that they contain a very large center for certain values of their parameters.

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The first named author acknowledges the financial support received from the Natural Sciences and Engineering Research Council of Canada via its Discovery Grant program. We are grateful to Pavel Etingof for useful suggestions and comments and to Valerio Toledano Laredo for numerous helpful discussions. We warmly thank the referee for a careful reading of our manuscript. Part of the work was done when the second named author visited the Mathematical Sciences Research Institute and the Max Planck Institute for Mathematics. She wishes to acknowledge the hospitality of MSRI and MPIM.

Received 16 November 2015

Accepted 18 July 2016

Published 29 December 2017