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Consider the following two similar situations.

  1. Let $G$ be an algebraic group (namely, of finite type) over a field $k,$ and let $\rho$ be a faithful representation of $G$ over $k.$ Then $\rho$ generates $Rep_k(G)$ as a Tannakian category.

  2. Let $G$ be a real compact Lie group, and let $\rho$ be a faithful complex representation of $G,$ with entries $g\mapsto a_{ij}(g).$ Then the $a_{ij}$'s and $\bar{a}_{ij}$'s generate the set of representative functions on $G$ (cf. Bröcker and tom Dieck, Representations of Compact Lie groups, GTM 98) as a $\mathbb C$-algebra.

Is there any reason for this similarity? I guess the name of Tannaka was initially involved in the second situation (reconstructing $G$ from its representative functions), and the fact that compact Lie groups (resp. representations of compact Lie groups) complexify to complex algebraic groups (resp. algebraic representations of complex algebraic groups) gives the correspondence between 1 (when the base field is the complex numbers) and 2. I would like to hear more comments from you.

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These statements are equivalent; you've made things more mysterious than they need to be by stating 1 in terms of the algebraic group instead of the compact group, where it's also true (and presumably easier to prove).

The equivalence follows immediately from the easy fact that:

The functions generated by matrix coefficients of V under multiplication are exactly the span of the matrix coefficients of all the tensor powers of V.

Thus, if your tensor powers contain all irreps, you get all representative functions. Like wise, if you get all representative functions, then you must have any irrep somewhere in a tensor power.

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A good reference is Procesi, Lie groups, Chapter 6, Cor 2.6 and Chapter 8, Theorem 3.2. –  Victor Protsak Oct 4 '10 at 6:30
    
Probably this kind of question first came up in Burnside's study of finite groups a century ago, "finite" being a very special type of "compact". –  Jim Humphreys Oct 4 '10 at 22:51
    
Dear Ben: I don't think they are quite equivalent (when k=C), because for a C-algebraic group to have a compact form, it is necessarily reductive. Also the rep are of different flavors: in 1, we consider rational rep (which is sort of algebraic), while in 2 we consider continuous (turns out to be analytic though...) ones. There seems to be a GAGA issue (which, of course, can be proved). –  shenghao Dec 18 '10 at 17:37
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