# Yang-Baxter endomorphisms

September 09, 2019

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.

Keywords:

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