Quantum algebras and quivers

Abstract

Given a finite quiver (Q) without loops, we introduce a new class of quantum algebras D(Q) which are deformations of the enveloping algebra of a Lie algebra which is a central extension of \(\mathfrak{sl}_n (\Pi(Q))\) where \(\Pi(Q)\) is the preprojective algebra of (Q). When Q is an affine Dynkin quiver of type A, D or E, we can relate them to Γ-deformed double current algebras. We are able to construct functors between different categories of modules over D(Q). We also give some general results about \(\widehat{\mathfrak{sl}}_n(A)\), for a quadratic algebra A and about \(\widehat{\mathfrak{g}}({\mathbb{C}}[u,v])\), which we use to introduce deformed double current algebras associated to a simple Lie algebra \(\mathfrak{g}\).