Shuffle approach towards quantum affine and toroidal algebras

TL;DR

This paper provides a comprehensive overview of shuffle algebra techniques applied to quantum affine and toroidal algebras, including new bases, geometric interpretations, and explicit computations of commutative subalgebras.

Contribution

It introduces shuffle algebra realizations for various quantum algebras, constructs new PBWD bases, and connects algebraic, geometric, and combinatorial perspectives.

Findings

Constructed new PBWD bases for quantum loop algebras.

Provided geometric interpretation of Fock modules.

Explicitly computed Bethe commutative subalgebras.

Abstract

These are detailed lecture notes of the crash-course on shuffle algebras delivered by the author at Tokyo University of Marine Science and Technology during the second week of March 2019. These notes consist of three chapters, providing a separate treatment for: the quantum loop algebras of $\mathfrak{sl}_n$ (as well as their super- and 2-parameter generalizations), the quantum toroidal algebras of $\mathfrak{gl}_1$, and the quantum toroidal algebras of $\mathfrak{sl}_n$. We provide the shuffle realization of the corresponding ``positive'' subalgebras as well as of the commutative subalgebras and some combinatorial representations for the toroidal algebras. One of the key techniques involved is that of ``specialization maps''. Each chapter aims to emphasize a different aspect of the theory: in the first chapter we use shuffle algebras to construct a family of new PBWD bases for type $A$ quantum loop algebras and their integral forms; in the second chapter, we provide a geometric interpretation of the Fock modules and use shuffle description of a commutative subalgebra to construct an action of the Heisenberg algebra on the equivariant $K$-theory of the Hilbert schemes of points; in the last chapter, we relate vertex and combinatorial representations of quantum toroidal algebras of $\mathfrak{sl}_n$ using Miki's isomorphism and use shuffle realization to explicitly compute Bethe commutative subalgebras and their limits. The latter construction is inspired by Enriquez's work relating shuffle algebras to the correlation functions of quantum affinized algebras.