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

## TL;DR

This paper constructs a family of projective unitary representations of the Thompson group F on Fermionic Fock space and investigates the conditions under which these can be lifted to true representations.

## Contribution

It introduces a new family of projective representations of F derived from CAR algebra and analyzes their liftability to genuine representations.

## Key findings

- Constructed projective representations on Fermionic Fock space.
- Computed the second cohomology group H^2(F;S^1) for these representations.
- Discussed conditions for lifting projective representations to ordinary ones.

## Abstract

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.