Some unitary representations of Thompson’s groups $F$ and $T$

Abstract

In a “naive” attempt to create algebraic quantum field theories on the circle, we obtain a family of unitary representations of Thompson’s groups $T$ and $F$ for any subfactor. The Thompson group elements are the “local scale transformations” of the theory. In a simple case the coefficients of the representations are polynomial invariants of links. We show that all links arise and introduce new “oriented” subgroups of $\overrightarrow{F}<F$ and $\overrightarrow{T}<T$ which allow us to produce all oriented knots and links.