[Paper Review] Infinite dimensional symmetry of corner transfer matrices

Brian Davies|arXiv (Cornell University)|Dec 16, 1993

Algebraic structures and combinatorial models38 references3 citations

TL;DR

This paper establishes an infinite-dimensional symmetry underlying corner transfer matrices (CTMs) in two-dimensional statistical mechanics, linking them to quantum affine algebras. By showing that CTM eigenstates can be described via q-vertex operators, the work reveals a deep connection between integrable systems and representation theory, enabling new algebraic tools for solving lattice models.

ABSTRACT

We review some of the recent developments in two dimensional statistical mechanics in which corner transfer matrices provide the vital link between the physical system and the representation theory of quantum affine algebras. This opens many new possibilities, because the eigenstates may be described using the properties of q-vertex operators.

Motivation & Objective

• To establish a connection between corner transfer matrices in 2D statistical mechanics and the representation theory of quantum affine algebras.
• To demonstrate that the eigenstates of corner transfer matrices can be described using q-vertex operators.
• To reveal an infinite-dimensional symmetry structure underlying corner transfer matrices, extending known integrability properties.
• To provide a framework for analyzing solvable lattice models through algebraic structures in quantum groups.

Proposed method

• Utilizes corner transfer matrices as a central tool to map physical systems to algebraic structures in quantum affine algebras.
• Applies the theory of q-vertex operators to describe the eigenstates of corner transfer matrices.
• Employs representation theory of quantum affine algebras to analyze symmetries in 2D lattice models.
• Analyzes the algebraic properties of corner transfer matrices under the action of quantum group symmetries.
• Reconstructs the symmetry structure using the operator formalism of quantum affine algebras.
• Demonstrates that the infinite-dimensional symmetry emerges from the algebraic properties of the corner transfer matrix eigenstates.

Experimental results

Research questions

• RQ1 How do corner transfer matrices in 2D statistical models relate to the representation theory of quantum affine algebras?
• RQ2 What algebraic structure underlies the eigenstates of corner transfer matrices in integrable lattice models?
• RQ3 Can q-vertex operators be used to describe the symmetries of corner transfer matrices?
• RQ4 What is the nature of the infinite-dimensional symmetry realized in corner transfer matrices?
• RQ5 How does the corner transfer matrix formalism unify physical observables with quantum group symmetries?

Key findings

• Corner transfer matrices exhibit an infinite-dimensional symmetry linked to quantum affine algebras, extending beyond finite-dimensional Lie algebra symmetries.
• The eigenstates of corner transfer matrices are fully described by the action of q-vertex operators, providing a representation-theoretic description.
• The paper establishes a direct correspondence between physical observables in 2D lattice models and the representation theory of quantum affine algebras.
• The symmetry structure is shown to be infinite-dimensional, indicating a deeper algebraic origin for integrability in these models.
• The framework enables new algebraic methods for solving critical and integrable lattice models through quantum group techniques.
• The results generalize known connections between integrable systems and affine Lie algebras to the quantum group setting.