Deformed Double Current Algebras via Deligne Categories

In this paper, we give an alternative construction of a certain class of deformed double current algebras. These algebras are deformations of U(End(k(r))[x,y]) and they were initially defined and studied by N. Guay in his papers. Here, we construct them as algebras of endomorphisms in Deligne category. We do this by taking an ultraproduct of spherical subalgebras of the extended Cherednik algebras of finite rank.