Work out the Tannakian duality for the closed subgroups

first as a consequence of the general results that we have, regarding the closed subgroups

and then independently, by pointing out the simplifications that appear in the real case.

Work out the Tannakian duality for the closed subgroups

whose fundamental representation is self-adjoint, up to equivalence,

first as a consequence of the results that we have, and then independently.

Check the Brauer theorems for [math]O_N,U_N[/math], which are both of type

for small values of the global length parameter, [math]k+l\in\{1,2,3\}[/math].

Write down Brauer theorems for the bistochastic groups

by identifying first the partition which produces them, as subgroups of [math]O_N,U_N[/math].

Look up the original version of Tannakian duality, stating that [math]G[/math] can be recovered from the knowledge of its full category of representations [math]\mathcal R_G[/math], viewed as subcategory of the category [math]\mathcal H[/math] of the finite dimensional Hilbert spaces, with each [math]\pi\in\mathcal R_G[/math] corresponding to its Hilbert space [math]H_\pi\in\mathcal H[/math], and write down a brief account of this.

Look up the Doplicher-Roberts and Deligne theorems, stating that the compact group [math]G[/math] can be in fact recovered from the sole knowledge of the category [math]\mathcal R_G[/math], with no need for the embedding into [math]\mathcal H[/math], and write down a brief account of this.

Given a closed subgroup [math]G\subset_uU_N[/math], understand and then briefly explain, in a short piece of writing, why the [math]*[/math]-algebras

form a planar algebra in the sense of Jones, and then comment as well on the various formulations of Tannakian duality, in the planar algebra setting.