Lechner, Gandalf  ORCID: https://orcid.org/0000-0002-8829-3121, Pennig, Ulrich ORCID: https://orcid.org/0000-0001-5441-6130 and Wood, Simon ORCID: https://orcid.org/0000-0002-8257-0378
2019.
Yang-Baxter representations of the infinite symmetric group.
Advances in Mathematics
355
, 106769.
10.1016/j.aim.2019.106769
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Abstract
Every unitary involutive solution of the quantum Yang-Baxter equation (“R-matrix”) defines an extremal character and a representation of the infinite symmetric group S∞. We give a complete classification of all such Yang-Baxter characters and determine which extremal characters of S∞ are of Yang-Baxter form. Calling two involutive R-matrices equivalent if they have the same character and the same dimension, we show that equivalence classes are classified by pairs of Young diagrams, and construct an explicit normal form R-matrix for each class. Using operator-algebraic techniques (subfactors), we prove that two R-matrices are equivalent if and only if they have similar partial traces. Furthermore, we describe the algebraic structure of the equivalence classes of all involutive R-matrices, and discuss several classes of examples. These include Yang-Baxter representations of the Temperley-Lieb algebra at parameter q=2, which can be completely classified in terms of their rank and dimension.
| Item Type: | Article |
|---|---|
| Date Type: | Publication |
| Status: | Published |
| Schools: | Schools > Mathematics |
| Subjects: | Q Science > QA Mathematics |
| Publisher: | Elsevier |
| ISSN: | 0001-8708 |
| Date of First Compliant Deposit: | 5 August 2019 |
| Date of Acceptance: | 5 August 2019 |
| Last Modified: | 13 Nov 2024 09:00 |
| URI: | https://orca.cardiff.ac.uk/id/eprint/124722 |
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