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.

References in corpus (26)

• Non-commutative Donaldson-Thomas theory and the conifold
• Brane Tilings and Their Applications
• Crystal Melting and Toric Calabi-Yau Manifolds
• Quiver Yangian from Crystal Melting
• Superintegrability for ($\beta$-deformed) partition function hierarchies with $W$-representations
• Superintegrability summary
• Quiver Yangian and Supersymmetric Quantum Mechanics
• Shifted Quiver Yangians and Representations from BPS Crystals
• Interpolating Matrix Models for WLZZ series
• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
• Many-body integrable systems implied by WLZZ models
• Gauge/Bethe correspondence from quiver BPS algebras
• Crystal Melting and Wall Crossing Phenomena
• A Note on Quiver Quantum Toroidal Algebra
• Braid actions on quantum toroidal superalgebras
• Super-Schur Polynomials for Affine Super Yangian $\mathsf{Y}({\widehat{\mathfrak{gl}}}_{1|1})$
• The Origin of Calabi-Yau Crystals in BPS States Counting
• 3-Schurs from explicit representation of Yangian $Y({\widehat{\mathfrak{gl}}}_1)$. Levels 1-5
• Simple Representations of BPS Algebras: the case of $Y({\widehat{\mathfrak{gl}}}_2)$
• More on Affine Dynkin Quiver Yangians
• Wall-Crossing Effects on Quiver BPS Algebras
• Charging solid partitions
• Algorithms for representations of quiver Yangian algebras
• Macdonald polynomials for super-partitions
• Crystals and Double Quiver Algebras from Jeffrey-Kirwan Residues
• An Overview of Crystals and Double Quiver Yangians