Quiver Yangian algebras associated to Dynkin diagrams of A-type and their rectangular representations

Abstract

The connection between simple Lie algebras and their Yangian algebras has a long history. In this work, we construct finite-dimensional representations of Yangian algebras $\mathsf{Y}(\mathfrak{sl}_{n})$ using the quiver approach. Starting from quivers associated to Dynkin diagrams of type A, we construct a family of quiver Yangians. We show that the quiver description of these algebras enables an effective construction of representations with a single non-zero Dynkin label. For these representations, we provide an explicit construction using the equivariant integration over the corresponding quiver moduli spaces. The resulting states admit a crystal description and can be identified with the Gelfand-Tsetlin bases for $\mathfrak{sl}_{n}$ algebras. Finally, we show that the resulting Yangians possess notable algebraic properties, and the algebras are isomorphic to their alternative description known as the second Drinfeld realization.