I learnt here that a) Reshitikhine-Turaev works with any ribbon category but
b) those not coming from Lie groups are rare.  

Can someone give an actual example (and preferrable with purely graphic definitions; a skein relation would be most handy) of the non-Lie variety? Just in case the Turaev book ("Quantum Invariants...") helps, my math library has it but some dude seems to be hamstering all copies :-) (So a scan would be necessary.)



Non Lie-group ribbon categories
I guess the OP refers to Reshetikhin-Turaev invariants coming from ribbon categories which are the representation category of quantum groups at root of unity. This construction works for every non-degenerated ribbon category, also called modular tensor category. (Non-degenerate means that the S-matrix is invertible. The S-matrix is obtained as follows, the entry $S\_{ij}$ is the categorical trace of a "double braid" $c\_j,i\circ cc\_i,j\in \mathrm{Hom}(i\otimes j,i\otimes j)$ where $i,j$ run over a choice of isomorphism classes of simple objects, in other words, $S\_{i,j}$ is the value of the Hopf link colored with $i$ and $j$.)


So I guess the OP is basically asking for modular tensor categories which are not coming from quantum groups.


Examples are Drinfeld centers of fusion categories, but this invariants coincide with the Turaev-Viro invariant for the fusion category and are in general not coming from q-deformations of Lie algebras/Quantum groups.


There are two examples of this form discussed in <http://arxiv.org/abs/0710.5761>, one comes from the even part of a $E\_6$ subfactor and the other from the even part of the Haagerup subfactor. I guess for both examples there is no skein theory established, and it is probably easier to calculate the invariants from the underlying fusion category.


There are also standard constructions to pass from one modular tensor category to another one by taking, for example, commutative algebra objects in a given category. In this case, the local modules of the algebra give a new modular tensor category.
There is discussion in
<http://arxiv.org/abs/1009.2117>


There is also a list of some known examples of modular tensor categories on <http://www.math.ksu.edu/~gerald/voas/>


