A homological model for Uq𝔰𝔩2 Verma modules and their braid representations

Martel, Jules

doi:10.2140/gt.2022.26.1225

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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

DOI: 10.2140/gt.2022.26.1225

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 $U_{q} 𝔰 𝔩 (2)$ . 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 $U_{q} 𝔰 𝔩 (2)$ , 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 $U_{q} 𝔰 𝔩 (2)$ –module.

Keywords

quantum groups, braid representations, Verma modules, configuration spaces, twisted homology

Mathematical Subject Classification 2010

Primary: 17B37, 20F36, 57M27, 57R56

Secondary: 55N25, 55R80, 57M10

References

Publication

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

Authors

Jules Martel
Max Planck Institute for Mathematics Bonn Germany
https://guests.mpim-bonn.mpg.de/martel/

Volume 26, issue 3 (2022)