A Family of Projective Representations of the Thompson Group and Lifting Problems

The Thompson group F has a natural unitary representation on $H=L^2[0,1]$. With some projections, we construct a family of projective unitary representations on a Fermionic Fock space associated with $H$. It comes from the representation of the associated CAR algebra. After $H^2(F;S^1)$ is obtained, we mainly study whether any of these projective unitary representations can be lifted to an ordinary one. We will discuss the lifting problem of these projective representations.