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

A. Gavshin

TL;DR

This work develops a quiver-theoretic construction of Yangian algebras $ extsf{Y}( rak{sl}_n)$ for A-type Dynkin diagrams and produces explicit finite-dimensional rectangular representations labeled by $(p,\lambda)$. Using quiver data, Jacobian algebras, and equivariant localization, the authors describe crystal representations whose states form Gelfand–Tsetlin bases, with actions of the Yangian generators realized by adding/removing atoms in a colored crystal. They demonstrate explicit constructions for $ extsf{Y}( rak{sl}_3)$ and $ extsf{Y}( rak{sl}_4)$ and extend the method to general $ extsf{Y}( rak{sl}_n)$, including the representations $epsilon_{p,\lambda}$, with detailed GT-pattern parametrizations and edge eigenfunctions. The results establish isomorphisms to Drinfeld’s second realization, reveal embedding structures among ranks, and connect quiver BPS algebras to classical representation theory, while outlining limitations and directions for extending to non-A diagrams and more general 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.

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

TL;DR

This work develops a quiver-theoretic construction of Yangian algebras for A-type Dynkin diagrams and produces explicit finite-dimensional rectangular representations labeled by . Using quiver data, Jacobian algebras, and equivariant localization, the authors describe crystal representations whose states form Gelfand–Tsetlin bases, with actions of the Yangian generators realized by adding/removing atoms in a colored crystal. They demonstrate explicit constructions for and and extend the method to general , including the representations , with detailed GT-pattern parametrizations and edge eigenfunctions. The results establish isomorphisms to Drinfeld’s second realization, reveal embedding structures among ranks, and connect quiver BPS algebras to classical representation theory, while outlining limitations and directions for extending to non-A diagrams and more general 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 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 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.

Paper Structure

This paper contains 25 sections, 192 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Quiver Yangian algebras
  3. Quiver Data
  4. Yangian Algebra
  5. Basic Properties
  6. States
  7. Yangian Representations
  8. Equivariant Matrix Coefficients
  9. Dynkin Diagrams
  10. Yangian Algebras $\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})$
  11. $\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{3})$ example
  12. The Algebra
  13. The States
  14. The Representations $\Upsilon_{1,\lambda}$
  15. $\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{4})$ example
  16. ...and 10 more sections

Figures (6)

  • Figure 1: Example of a quiver
  • Figure 2: The quiver that describes $(\lambda, 0)$ reps of $\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{3})$
  • Figure 3: $\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{4})$ quivers
  • Figure 4: The structure of the representation $\Upsilon_{2, 1}$ of the algebra $\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{4})$
  • Figure 5: Examples of inadmissible empty boxes
  • ...and 1 more figures