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 2 There 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