A homological model for Uqsl2 Verma modules and their braid representations

Abstract

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