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 $\mathfrak{g}$.