This book is based on the author's mini course delivered at Tokyo University of Marine Science
and Technology in March 2019. The shuffle approach to Drinfeld-Jimbo quantum groups of finite
type (embedding their positive subalgebras into q-deformed shuffle algebras) was first
developed independently in the 1990s by J. Green M. Rosso and P. Schauenburg. Motivated by
similar ideas B. Feigin and A. Odesskii proposed a shuffle approach to elliptic quantum groups
around the same time. The shuffle algebras in the present book can be viewed as trigonometric
degenerations of the Feigin-Odesskii elliptic shuffle algebras. They provide combinatorial
models for the positive subalgebras of quantum affine algebras in their loop realizations.
These algebras appeared first in that context in the work of B. Enriquez. Over the last decade
the shuffle approach has been applied to various problems in combinatorics (combinatorics of
Macdonald polynomials and Dyck paths generalization to wreath Macdonald polynomials and
operators) geometric representation theory (especially the study of quantum algebras' actions
on the equivariant K-theories of various moduli spaces such as affine Laumon spaces Nakajima
quiver varieties nested Hilbert schemes) and mathematical physics (the Bethe ansatz quantum
Q-systems and quantized Coulomb branches of quiver gauge theories to name just a few). While
this area is still under active investigation the present book focuses on quantum affine
toroidal algebras of type A and their shuffle realization which have already illustrated a
broad spectrum of techniques. The basic results and structures discussed in the book are of
crucial importance for studying intrinsic properties of quantum affinized algebras and are
instrumental to the aforementioned applications.
