** Representation theory and cohomology for standard invariants **

We study pairs of factors, with finite.

There are a bunch of discrete invariants.

- Standard invariants
- Rigid algebras
- Planar algebras (Jones).

How can these invariants act on factors ? This is Popa’s theory.

We shall introduce representations on Hilbert spaces, and define Betti numbers.

**1. Overview: standard invariants **

** 1.1. Jones tower **

With a pair , there comes a whole tower , with equal indices. . is a grid of multimatrix algebras with if ). Popa calls such a structure a -lattice.

** 1.2. Alternative view **

View as an -bimodule. By tensor products over , one gets more modules . Then, as an -bimodule, . Also, . We get a 2-category (of -bimodules) with generator .

** 1.3. Planar algebras **

This is a diagrammatic way to write intertwiners .

** 1.4. Popa’s classification **

Theorem 1 (Popa 1992)The standard invariant is a complete invariant of the pair when is the hyperfinite factor and the standard invariant is amenable.

This suggests to define amenability, property (T), Haagerup property,… for the standard invariants. This was done by Popa. In 2014, Popa and I gave an intrinsic definition for these notions fir -lattices and tensor categories. Very soon, Neshveyev-Yamashita defined unitary representations (the Drinfeld center). Immediately after, Ghosh-C. Jones showed that unitary representations coincide with ordinary representations of the tube algebra.

**2. The symmetric-enveloping inclusion **

Now we start again with a more synthetic view.

Start with . Let . Let be the unique factor generated by and a projection and such that

- are the toric construction for .
- Idem for .

This automatically contains the entire Jones tower and also the Jones tunnel (a tower contained in ). Then

Longo and Rehren, Morsodo have an alternative construction for . Consider the category of all finite index -bimodules appearing in . Every irredcible is realised by . Define

This has an obvious product.

Theorem 2There is an involution and a positive functional ,

Hence we get a pair of von Neumann algebras . is a trace when is extremal (meaning that all bimodules in have equal left and right dimension.

**3. Quasiregular inclusions **

is called quasiregular if belongs to the closure of the span of finite index subbimodules.

**Example 1**. The symmetric-enveloping inclusion just described is.

**Example 2**. Crossed products .

The more general approach (by Popa, Shlyakhtenko and myself) goes as follows.

A “unitary respresentation” for is a Hilbert -bimodule such that as a -bimodule , we have a direct sum of -bimodules in a given category .

A “positive definite function” is a completely positive -linear map . In the symmetric-enveloping example, such maps are scalar on each , therefore it boils down to a -valude fonction on . There in an obvious DNS construction that passes from positive definite functions to unitary representations.

**4. Plan **

In lecture 2, I will attach to an -algebra (like ).

In lecture 3, I will do cohomology, -Betti numbers (like ).

In lecture 4, I will give examples and computations.