This is a question about Lusztig's theory of based modules. This theory is elementary but far from easy and is developed in Chapter 27 of "Introduction to quantum groups".
Let $V$ be a highest weight representation of a quantum group. Then it is well-known that the r-string braid group acts on the space of (co)invariant tensors in the r-th tensor power of $V$. Using the theory of based modules Lusztig constructs a basis of this vector space. I would like to know if it is possible to calculate the action of the braid group with respect to this basis, (or the dual basis)?
The naive approach is to:
1) calculate the braid group action on the tensor product basis
2) calculate the Lusztig basis in terms of the tensor product basis
3) deduce the desired action
This approach is clearly complicated and is also impractical. I was hoping for a more direct method. The reason I am hopeful is that the quasi R-matrix is used in the definition of the tensor product of based modules and the R-matrix differs from the R-matrix only in diagonal terms.
Also the case SL(2) is known. This is in a paper by Khovanov and Frenkel.
I am particularly interested in quasi-miniscule representations and in these cases it is straightforward to give the action of the quantum group with respect to the canonical basis. This is done in Jantzen's book.