Yang-Baxter endomorphisms

TL;DR

This paper provides a comprehensive study of Yang-Baxter endomorphisms, exploring their algebraic, spectral, and index properties, and their connections to braid group characters and R-matrix invariants.

Contribution

It introduces new characterizations of Yang-Baxter endomorphisms, analyzes their subfactor structures, and fully classifies all such endomorphisms in dimension two.

Findings

Partial trace of R-matrix is an invariant for its character.

Left and right partial traces of R-matrices coincide and are normal.

Spectrum of R-matrix cannot be concentrated in a small disc.

Abstract

Every unitary solution of the Yang-Baxter equation (R-matrix) in dimension $d$ can be viewed as a unitary element of the Cuntz algebra ${\mathcal O}_d$ and as such defines an endomorphism of ${\mathcal O}_d$. These Yang-Baxter endomorphisms restrict and extend to endomorphisms of several other $C^*$- and von Neumann algebras and furthermore define a II$_1$ factor associated with an extremal character of the infinite braid group. This paper is devoted to a detailed study of such Yang-Baxter endomorphisms. Among the topics discussed are characterizations of Yang-Baxter endomorphisms and the relative commutants of the various subfactors they induce, an endomorphism perspective on algebraic operations on R-matrices such as tensor products and cabling powers, and properties of characters of the infinite braid group defined by R-matrices. In particular, it is proven that the partial trace of an R-matrix is an invariant for its character by a commuting square argument. Yang-Baxter endomorphisms also supply information on R-matrices themselves, for example it is shown that the left and right partial traces of an R-matrix coincide and are normal, and that the spectrum of an R-matrix can not be concentrated in a small disc. Upper and lower bounds on the minimal and Jones indices of Yang-Baxter endomorphisms are derived, and a full characterization of R-matrices defining ergodic endomorphisms is given. As examples, so-called simple R-matrices are discussed in any dimension $d$, and the set of all Yang-Baxter endomorphisms in $d=2$ is completely analyzed.