## Notes of Stefan Vaes’ second Cambridge lecture 18-01-2017

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 ${II_1}$ factors ${T\subset S}$ is quasiregular if ${S}$ is spanned by finite index ${T}$-submodules.

The main examples are SE-inclusions and cross-products ${T\times\Gamma}$ attached to outer actions.

The definition has algebraic content: there is a canonical dense ${*}$-subalgebra ${\zeta}$, the linear span finite index ${T}$-submodules. In cross-product case, it is spanned by ${Tu_g}$, ${g\in\Gamma}$.

Theorem 1 (Popa-Schlyakhtenko-Vaes, after people did it in special cases) We construct a ${*}$-algebra ${\mathcal{A}}$ such that its Hilbert representations are in ${1-1}$ correspondence with ${S}$-bimodules

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

1.1. Construction of ${\mathcal{A}}$

Data: ${T\subset S}$ and a category ${\mathcal{C}}$ of finite index ${T}$-bimodules.

As a vectorspace,

$\displaystyle \mathcal{A}=\bigoplus_{i,j\in Irr(\mathcal{C})}(i\zeta,\zeta j),$

where ${(i\zeta,\zeta j)}$ denotes the space of finite rank intertwiners from ${L^2(S)\otimes_T H_j}$ to ${H_i \otimes_T L^2(S)}$. Finite rank means that it involves only finitely many of the finite index ${T}$-modules constituting ${S}$.

Notation. Given ${\mathcal{F}}$ a subset of irreducibles, denote by ${e_{\mathcal{F}}:L^2(S)\rightarrow L^2(S)}$ the projections onto the span of all of those in ${\mathcal{F}}$.

Then an intertwiner ${V}$ is finite rank if ${V=V(e_{\mathcal{F}}\otimes 1)=(1\otimes e_{\mathcal{F}}V}$ for some finite subset ${\mathcal{F}}$ of irreducibles.

The composition of intertwiners is defined by

$\displaystyle \begin{array}{rcl} VW=(1\otimes m)(V\otimes 1)(1\otimes W)(m^*\otimes 1), \end{array}$

where ${m:L^2(S)\otimes L^2(S)\rightarrow L^2(S)}$ is induced by multiplication.

There are canonical idempotents ${p_i\in\mathcal{A}}$, ${i\in Irr(\mathcal{C})}$, such that ${p_i\mathcal{A}p_j=(i\zeta,\zeta j)}$. The inclusion defines an element ${\delta:L^2(T)\rightarrow L^2(S)}$ such that ${p_i=(1\otimes \delta)(\delta^*\otimes 1)\in (i\zeta,\zeta i)}$.

The adjoint is defined formally by ${t:L^2(S)\rightarrow L^2(S)\otimes_T L^2(S)}$ as ${t=m^*\delta}$. In fact, only ${(1\otimes e_{\mathcal{F}})t}$ makes sense. In reality, define, for ${V\in (i\zeta,\zeta j)}$,

$\displaystyle \begin{array}{rcl} V^\#=(t^*\otimes 1\otimes 1)(1\otimes V^*\otimes 1)(1\otimes 1\otimes t). \end{array}$

This turns ${\mathcal{A}}$ into a ${*}$-algebra.

1.2. Example of cross-product

Let ${S=T\times \Gamma}$. Then ${\zeta=\bigoplus_{g\in \Gamma}g}$ is the algebraic direct sum of irreducible ${T}$-bimodules. ${(g\zeta,\zeta h)=\bigoplus_{k,k'}(gk,k'h)}$, where each summand is ${{\mathbb C}}$ if ${g=k'gk^{-1}}$, 0 otherwise. Thus ${\mathcal{A}}$ is built from the action of ${\Gamma}$ on its set of conjugacy classes,

$\displaystyle \begin{array}{rcl} \mathcal{A}=C_c(\Gamma)\times\Gamma. \end{array}$

Also, the group algebra ${{\mathbb C}\Gamma=p_e\mathcal{A}p_e}$ is merely a corner in ${\mathcal{A}}$.

Theorem 2 There is a ${1-1}$ correspondence between Hilbert ${S}$-bimodules ${H}$ such that ${_T H_T}$ is a sum of objects of ${\mathcal{C}}$, and non-degenerate right Hilbert ${\mathcal{A}}$-modules ${K}$. The correspondence is

$\displaystyle \begin{array}{rcl} \forall i\in Irr(\mathcal{C}),\quad Kp_i=(H,H_i). \end{array}$

2. Construction of Ocneanu’s tube algebra

Data: a rigid ${C^*}$-tensor category ${\mathcal{C}}$.

As a vectorspace,

$\displaystyle \begin{array}{rcl} \mathcal{A}=\bigoplus_{i,j\in Irr(\mathcal{C})}\bigoplus_{\alpha\in Irr(\mathcal{C})}(i\alpha,\alpha j), \end{array}$

There are canonical idempotents ${p_i\in\mathcal{A}}$, ${i\in Irr(\mathcal{C})}$, such that

$\displaystyle p_i\mathcal{A}p_j=\bigoplus_{\alpha\in Irr(\mathcal{C})}(i\alpha,\alpha j),$

and ${p_E}$ such that

$\displaystyle \begin{array}{rcl} p_E\mathcal{A}p_E=\bigoplus_{\alpha\in Irr(\mathcal{C})}{\mathbb C}, \end{array}$

which I denote by ${{\mathbb C}[Irr(\mathcal{C})]}$. I call possitive the maps ${\omega:{\mathbb C}[Irr(\mathcal{C})]\rightarrow{\mathbb C}}$ such that

$\displaystyle \begin{array}{rcl} \forall V\in p_E \mathcal{A},\quad \omega(VV^\#)\geq 0. \end{array}$

The multiplication of ${V\in (i\alpha,\alpha j)}$ and ${W\in (j\alpha,\alpha k)}$ is

$\displaystyle \begin{array}{rcl} VW=\bigoplus_{\gamma\in Irr(\mathcal{C})}(\bigoplus_{X\in(\alpha\beta,\gamma)}(1\otimes X)(V\otimes 1)(1\otimes W)(X\otimes 1)). \end{array}$

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 ${X}$,and for each ${\alpha\in\mathcal{C}}$, a unitary morphism ${\sigma_\alpha:\alpha X\rightarrow X\alpha}$ such that

$\displaystyle \begin{array}{rcl} (1\otimes V)\sigma_\alpha=\sigma_\beta (V\otimes 1) \end{array}$

if ${V\in (\beta,\alpha)}$, and

$\displaystyle \begin{array}{rcl} \sigma_{\alpha\beta}=(\sigma_\alpha\otimes 1)(1\otimes \sigma_\beta). \end{array}$

2.2. How to take tensor products of representations ?

OK in half-brading and bimodule pictures