# A faithful representation “generates” all the other representations

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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