If you are happy to work with the Operator Algebra approach to quantum groups, then Woronowicz's definition of a <a href="http://en.wikipedia.org/wiki/Compact_quantum_group">Compact Quantum Group</a> admits a perfect analogue of the Peter-Weyl theorem (looking at unitary corepresentations).  The Wikipedia page doesn't give any references, but a good (and freely available) survey is: <a href="http://arxiv.org/abs/math/9803122">arXiv:math/9803122v1 [math.FA]</a>.

But maybe this isn't what you're after...