A homological model for Uq

We extend Lawrence's representations of the braid groups to relative homology modules, and we show that they are free modules over a Laurent polynomials ring. define homological operators actually provide representation for an integral version $U_q \mathfrak{sl}(2)$. suggest isomorphism between given basis standard tensor products Verma it preserve ring coefficients, action \mathfrak{sl}(2)$, group their grading. This recovers Kohno's theorem relating absolute Lawrence with quantum on highest weight vectors. It is extension latter as get rid generic conditions parameters, recover entire product Verma-modules \mathfrak{sl}(2)$-module.