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I learnt that the Coxeter groups have a few members more than the classic simple Lie groups: $H_3, H_4$ and $I_2(p)$. Is there a Reshetikhin-Turaev invariant for those, too? If not, where does the construction fail (maybe there is not even an associated quantum group)?

BTW, you would already help me by filling out the gap:
$I_2(6)=G_2: 7\bigotimes7=1\bigoplus7\bigoplus14\bigoplus27$
$I_2(5): R\bigotimes R=...\bigoplus ...$ (also for the defining irrep R)
$I_2(4)=BC_2: 5\bigotimes5=1\bigoplus10\bigoplus14$

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Concerning the parenthetic question, there really isn't a natural quantum group with such a Weyl group (as far as I know). Even so, the non-crystallographic finite Coxeter groups are an interesting laboratory for a number of combinatorial steps in Lusztig's work on finite groups of Lie type: some of the combinatorics still makes sense even for $H_3, H_4$ etc. Concerning the "defining irrep" R for the dihedral group of order 10, what is meant? For other types you are dealing with irreps of the corresponding Lie group, though $BC_2$ gives a smaller choice of dimension 4. –  Jim Humphreys Nov 21 '11 at 14:34

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