#### Volume 26, issue 3 (2022)

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A homological model for $U_q\mathfrak{sl}2$ Verma modules and their braid representations

### Jules Martel

Geometry & Topology 26 (2022) 1225–1289
##### 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 ${U}_{q}\mathfrak{𝔰}\mathfrak{𝔩}\left(2\right)$. 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 ${U}_{q}\mathfrak{𝔰}\mathfrak{𝔩}\left(2\right)$, 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 ${U}_{q}\mathfrak{𝔰}\mathfrak{𝔩}\left(2\right)$–module.

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##### Keywords
quantum groups, braid representations, Verma modules, configuration spaces, twisted homology
##### Mathematical Subject Classification 2010
Primary: 17B37, 20F36, 57M27, 57R56
Secondary: 55N25, 55R80, 57M10