Compact groups are pretty much the same as complex reductive groups, but one must replace finite dimensional unitary representations with finite dimensional holomorphic represententations. The Weierstrass approximation theorem says that polynomials are dense in $L^2$ for compact spaces. Putting this together, this yields an algebraic Peter-Weyl theorem that polynomial functions on, say, $GL_n(C)$ are the direct sum of endomorphisms of the irreducible holomorphic representations. One can also prove this directly, by essentially the same methods as the Peter-Weyl theorem, but rather easier.

If you are interested in non-compact groups, then this may not be the answer you're looking for, but if you're interested in quantum groups, it is a reasonable starting point.

Compact groups are pretty much the same as complex reductive groups, but one must replace finite dimensional unitary representations with finite dimensional holomorphic representations. The Weierstrass approximation theorem says that polynomials are dense in $L^2$ for compact spaces. Putting this together, this yields an algebraic Peter-Weyl theorem that polynomial functions on, say, $GL_n(C)$ are the direct sum of endomorphisms of the irreducible holomorphic representations. One can also prove this directly, by essentially the same methods as the Peter-Weyl theorem, but rather easier.

If you are interested in non-compact groups, then this may not be the answer you're looking for, but if you're interested in quantum groups, it is a reasonable starting point.