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


About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
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