Jones representations of Thompson's group $F$ arising from Temperley-Lieb-Jones algebras

Abstract

Following a procedure due to V. Jones, using suitably normalized elements in a Temperley-Lieb-Jones (planar) algebra we introduce a 3-parametric family of unitary representations of the Thompson's group $F$ equipped with canonical (vacuum) vectors and study some of their properties. In particular, we discuss the behaviour at infinity of their matrix coefficients, thus showing that these representations do not contain any finite type component. We then focus on a particular representation known to be quasi-regular and irreducible and show that it is inequivalent to itself once composed with a classical automorphism of F. This allows us to distinguish three equivalence classes in our family. Finally, we investigate a family of stabilizer subgroups of $F$ indexed by subfactor Jones indices that are described in terms of the chromatic polynomial. In contrast to the first non-trivial index value for which the corresponding subgroup is isomorphic to the Brown-Thompson's group $F_3$, we show that when the index is large enough this subgroup is always trivial.

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References in corpus (4)

• A no-go theorem for the continuum limit of a periodic quantum spin chain
• The Homflypt polynomial and the oriented Thompson group
• On Jones Subgroup of R. Thompson's Group $T$
• Pythagorean representations of Thompson's groups

Cited by in corpus (9)

• Arborescence of positive Thompson links
• On the $3$-colorable subgroup $\mathcal{F}$ and maximal subgroups of Thompson's group $F$
• Irreducible Pythagorean representations of R. Thompson's groups and of the Cuntz algebra
• Positive oriented Thompson links
• An introduction to Thompson knot theory and to Jones subgroups
• On Jones' connections between subfactors, conformal field theory, Thompson's groups and knots
• Remarks on some maximal subgroups of $F$ and on the $\vec{F}$-index of knots
• The planar $3$-colorable subgroup $\mathcal{E}$ of Thompson's group $F$ and its even part
• An extension of Krishnan's central limit theorem to the Brown-Thompson groups