Yang-Baxter representations of the infinite symmetric group

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 $S_{\infty }$. We give a complete classification of all such Yang-Baxter characters and determine which extremal characters of $S_{\infty }$ 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 unitary Yang-Baxter representations of the Temperley-Lieb algebra at loop parameter $\delta =2$, which can be completely classified in terms of their rank and dimension.

36 pages, 1 figure. Slightly revised in order to comply with accepted journal version

References in corpus (4)

• A no-go theorem for the continuum limit of a periodic quantum spin chain
• Noncommutative independence from the braid group $B_{\infty }$
• Noncommutative Independence from Characters of the Infinite Symmetric Group $s_{\infty }$
• Noncommutative products of Euclidean spaces

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• Yang-Baxter endomorphisms
• On orthogonal projections related to representations of the Hecke algebra on a tensor space
• Fock representations of Zamolodchikov algebras and R-matrices