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Yang-Baxter endomorphisms

Conti, Roberto and Lechner, Gandalf 2020. Yang-Baxter endomorphisms. Journal of the London Mathematical Society 103 (2) , pp. 633-671. 10.1112/jlms.12387

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Every unitary solution of the Yang-Baxter equation (R-matrix) in dimension $d$ can be viewed as a unitary element of the Cuntz algebra $\CO_d$ and as such defines an endomorphism of $\CO_d$. 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.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Additional Information: This is an open access article under the terms of the Creative Commons Attribution‐NonCommercial‐NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non‐commercial and no modifications or adaptations are made.
Publisher: Wiley
ISSN: 0024-6107
Date of First Compliant Deposit: 2 September 2020
Date of Acceptance: 28 August 2020
Last Modified: 04 Mar 2021 15:36

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