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The classification problem for unitary R-Matrices with two eigenvalues

Gandalf Lechner

Abstract

The problem of classifying all unitary R-matrices of arbitrary finite dimension that have precisely two distinct eigenvalues is described, working up to a natural equivalence relation given by the characters of their braid group representations. Up to one class that might or might not exist in even dimension larger than two, a full classification theorem is obtained.

The classification problem for unitary R-Matrices with two eigenvalues

Abstract

The problem of classifying all unitary R-matrices of arbitrary finite dimension that have precisely two distinct eigenvalues is described, working up to a natural equivalence relation given by the characters of their braid group representations. Up to one class that might or might not exist in even dimension larger than two, a full classification theorem is obtained.
Paper Structure (3 sections, 11 theorems, 50 equations)

This paper contains 3 sections, 11 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. R-matrices, subfactors, and Markov traces
  3. Unitary Hecke R-matrices

Key Result

Proposition 1.2

Every unitary R-matrix $R$ with dimension $\dim R=2$ is equivalent to precisely one of the following matricesHere, and only in this proposition and its proof, the notation $R_1,\ldots,R_4$ refers to the specified matrices and not to eq:Rk. (with the usual identification $M_2\otimes M_2\cong M_4$): with uniquely fixed parameters $q,p$ or unordered parameter pairs $\{p,s\}$.

Theorems & Definitions (23)

  • Definition 1.1
  • Proposition 1.2
  • proof
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 13 more