Moduli of representations of R. Thompson’s groups and of the Cuntz algebra

Vaughan Jones recently discovered a remarkable connection between conformal theory, knot theory, and the Richard Thompson groups. This led to a powerful technology for constructing actions of the Thompson groups. One application was by Brothier and Jones where from a representation of the Pythagorean algebra P, a linear isometry from a Hilbert space to its direct sum square, they constructed the so-called Pythagorean unitary representation (in short, P-representation) of the Thompson groups. These representations extend to *-representations of the Cuntz algebra O, and all *-representations arise in this way, though not all representations of the Thompson groups are Pythagorean. We initiate a detailed study of these P-representations. The core of all our results is that remarkably rich information about these representations can be gleaned by studying representations of P. This converts difficult questions concerning infinite-dimensional spaces and operator algebras into much simpler ones in finite-dimensional linear algebra. We first introduce the Pythagorean dimension; a new invariant for all representations of O and all P-representations of the Thompson groups. We then define the notion of atomic and diffuse P-representations. Each P-representation decomposes as a direct sum of an atomic and a diffuse one. We completely describe the finite P-dimensional atomics as a direct sum of induced representations arising from certain one-dimensional representations. We then obtain a sharp dichotomy whereby we show that diffuse P-representations are Ind-mixing - they do not contain the induction of any finite-dimensional representation of any subgroup. Thus, this makes the diffuse P-representations the first of their kind, and highly non-trivial to construct (without using Jones’ technology in which we do not see an easier way to construct). We then classify the irreducible classes of finite P-dimensional diffuse P-representations by showing it is equivalent to classifying finite-dimensional representations of P. Subsequently, we obtain that the moduli space of these irreducible classes can be endowed with a rich geometrical structure. For each finite Pythagorean dimension d, the irreducible classes form a manifold of dimension 2d^2 + 1 and the atomic ones form a sub-manifold of dimension one. Hence, the irreducible classes of diffuse ones form a smooth manifold of dimension 2d^2 + 1.