** Representation theory and cohomology for standard invariants **

**1. Tube algebra of a quasiregular inclusion **

The historical reason for the word tube will not appear…

Recall that a finite index pair of factors is quasiregular if is spanned by finite index -submodules.

The main examples are SE-inclusions and cross-products attached to outer actions.

The definition has algebraic content: there is a canonical dense -subalgebra , the linear span finite index -submodules. In cross-product case, it is spanned by , .

Theorem 1 (Popa-Schlyakhtenko-Vaes, after people did it in special cases)We construct a -algebra such that its Hilbert representations are in correspondence with -bimodules

I explain this because it makes the tube algebra natural. Then I specialize to tensor categories.

** 1.1. Construction of **

Data: and a category of finite index -bimodules.

As a vectorspace,

where denotes the space of finite rank intertwiners from to . Finite rank means that it involves only finitely many of the finite index -modules constituting .

**Notation**. Given a subset of irreducibles, denote by the projections onto the span of all of those in .

Then an intertwiner is finite rank if for some finite subset of irreducibles.

The composition of intertwiners is defined by

where is induced by multiplication.

There are canonical idempotents , , such that . The inclusion defines an element such that .

The adjoint is defined formally by as . In fact, only makes sense. In reality, define, for ,

This turns into a -algebra.

** 1.2. Example of cross-product **

Let . Then is the algebraic direct sum of irreducible -bimodules. , where each summand is if , 0 otherwise. Thus is built from the action of on its set of conjugacy classes,

Also, the group algebra is merely a corner in .

Theorem 2There is a correspondence between Hilbert -bimodules such that is a sum of objects of , and non-degenerate right Hilbert -modules . The correspondence is

**2. Construction of Ocneanu’s tube algebra **

Data: a rigid -tensor category .

As a vectorspace,

There are canonical idempotents , , such that

and such that

which I denote by . I call possitive the maps such that

The multiplication of and is

We need to play with three equivalent points of view: the tube algebra, bimodules, and unitary half braidings.

For SE-inclusions, the equivalence of tube algebra and Bimodules is due to Ghosh-C. Jones.

** 2.1. Unitary half-braiding **

Data: an irreducible ,and for each , a unitary morphism such that

if , and

** 2.2. How to take tensor products of representations ? **

OK in half-brading and bimodule pictures