Yang-Baxter algebras as convolution algebras: The Grassmannian case

TL;DR

This paper links quantum integrable models with Schubert calculus by explicitly constructing solutions to the Yang-Baxter equation using geometric methods related to Grassmannians, revealing new algebraic structures.

Contribution

It provides a geometric construction of Yang-Baxter solutions as convolution algebras from equivariant Schubert calculus, connecting integrable models with algebraic geometry.

Findings

Identified Yang-Baxter algebra as convolution algebra from Grassmannian Schubert calculus

Constructed quotients of current algebra rak{gl}_2[t] acting on tensor products

Connected the approach with the Cohomological Hall algebra (COHA) for the A_1-quiver

Abstract

We present a simple but explicit example of a recent development which connects quantum integrable models with Schubert calculus: there is a purely geometric construction of solutions to the Yang-Baxter equation and their associated Yang-Baxter algebras which play a central role in quantum integrable systems and exactly solvable lattice models in statistical physics. We consider the degenerate five-vertex limit of the asymmetric six-vertex model and identify its associated Yang-Baxter algebra as convolution algebra arising from the equivariant Schubert calculus of Grassmannians. We show how our method can be used to construct (Schur algebra type) quotients of the current algebra $\mathfrak{gl}_2[t]$ acting on the tensor product of copies of its evaluation representation $\mathbb{C}^2[t]$. Finally we connect it with the COHA for the $A_1$-quiver.