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On the structure of tensor products of the finite-dimensional representations of quantum affine $\mathfrak{sl}_n$

Henrik Juergens

TL;DR

This work surveys the finite-dimensional representation theory of untwisted quantum affine algebras, with a focus on $U_q(\tilde{\mathfrak{sl}}_n)$, Drinfeld polynomials, and $q$-characters. It builds a foundational bridge from ordinary Lie theory to the loop/Drinfeld realizations, emphasizing braided monoidal categories and the role of the universal (pseudo) $R$-matrix in relating representations. A central theme is the introduction and study of snake modules, their $q$-character path formula, and the extended $\mathcal{T}$-system, which connects to cluster-algebra structures in the Grothendieck ring. The discussion also highlights evaluation representations via Jimbo’s homomorphism, the trigonometric $R$-matrix, and the physical interpretation of transfer matrices in integrable models, thereby linking algebraic structure to statistical-mechanical contexts and potential applications to correlation functions in higher-rank systems.

Abstract

We review some important facts about the structure of tensor products of finite dimensional representations of quantum affine algebras. This is done from the elementary standpoint of the representation theory of semisimple Lie algebras in the scope of a masters thesis. We set the focus on drawing the line to the finite dimensional representation theory of quantum affine $\mathfrak{sl}_n$, the theory of q-characters, and introduce the so called snake modules together with their character formula.

On the structure of tensor products of the finite-dimensional representations of quantum affine $\mathfrak{sl}_n$

TL;DR

This work surveys the finite-dimensional representation theory of untwisted quantum affine algebras, with a focus on , Drinfeld polynomials, and -characters. It builds a foundational bridge from ordinary Lie theory to the loop/Drinfeld realizations, emphasizing braided monoidal categories and the role of the universal (pseudo) -matrix in relating representations. A central theme is the introduction and study of snake modules, their -character path formula, and the extended -system, which connects to cluster-algebra structures in the Grothendieck ring. The discussion also highlights evaluation representations via Jimbo’s homomorphism, the trigonometric -matrix, and the physical interpretation of transfer matrices in integrable models, thereby linking algebraic structure to statistical-mechanical contexts and potential applications to correlation functions in higher-rank systems.

Abstract

We review some important facts about the structure of tensor products of finite dimensional representations of quantum affine algebras. This is done from the elementary standpoint of the representation theory of semisimple Lie algebras in the scope of a masters thesis. We set the focus on drawing the line to the finite dimensional representation theory of quantum affine , the theory of q-characters, and introduce the so called snake modules together with their character formula.
Paper Structure (29 sections, 90 theorems, 254 equations, 20 figures)

This paper contains 29 sections, 90 theorems, 254 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Structure
  3. Mathematical background
  4. Kac--Moody algebras
  5. The Kac--Moody algebras $L(A)$ and $L(A)'$
  6. GCM of finite and affine type
  7. The standard invariant bilinear form
  8. Kac--Moody algebras of affine type
  9. The loop construction for untwisted affine Kac--Moody algebras
  10. Cartan data of finite semisimple Lie algebras
  11. Quantum groups and quantum affine algebras
  12. Hopf algebras in terms of string diagrams
  13. Quasitriangular Hopf algebras
  14. Abelian monoidal categories
  15. Quantized universal enveloping algebras
  16. ...and 14 more sections

Key Result

Proposition 2.0.0.3

Let $A$ be any associative algebra with $1$ over $k$ and $[A]$ the corresponding Lie algebra. Then for any Lie algebra homomorphism $\theta:\mathfrak{g}\to [A]$ there exists a unique associative algebra homomorphism $\phi: U(\mathfrak{g})\to A$ such that $\phi\circ\sigma = \theta$, where $\sigma$ is

Figures (20)

  • Figure 1: A simple example of the extended $\operatorname{T}$-system for a minimal snake $(i_t,k_t)_{1\leq t\leq 5}$ in $A_4$. It is a short exact sequence of snake modules defined by the minimal snake, its neighbouring snakes and the snakes obtained from it by omitting the first or the last element.
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 15 more figures

Theorems & Definitions (193)

  • Definition 2.0.0.1: Lie algebra
  • Definition 2.0.0.2: universal enveloping algebra
  • Proposition 2.0.0.3
  • Proposition 2.0.0.4
  • Theorem 2.0.0.5: Poincaré-Birkhoff-Witt
  • Proposition 2.1.1.1
  • Definition 2.1.1.2
  • Proposition 2.1.1.3
  • Definition 2.1.1.4: the Lie algebra $\tilde{L}(A)$
  • Lemma 2.1.1.5
  • ...and 183 more