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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

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 Uq𝔰𝔩(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 Uq𝔰𝔩(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 Uq𝔰𝔩(2)–module.

quantum groups, braid representations, Verma modules, configuration spaces, twisted homology
Mathematical Subject Classification 2010
Primary: 17B37, 20F36, 57M27, 57R56
Secondary: 55N25, 55R80, 57M10
Received: 12 March 2020
Revised: 10 December 2020
Accepted: 10 February 2021
Published: 3 August 2022
Proposed: Ciprian Manolescu
Seconded: Ulrike Tillmann, András I Stipsicz
Jules Martel
Max Planck Institute for Mathematics