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A
homological model for $U_q\mathfrak{sl}2$ Verma modules and
their braid representations
Jules Martel |
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Geometry & Topology 26 (2022) 1225–1289
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Abstract |
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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 . 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 , 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 –module. |
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Keywords
quantum groups, braid representations, Verma modules,
configuration spaces, twisted homology
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Mathematical Subject Classification 2010
Primary: 17B37, 20F36, 57M27, 57R56
Secondary: 55N25, 55R80, 57M10
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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
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Authors |
| Jules Martel | |
| Max Planck Institute for
Mathematics Bonn Germany |
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| https://guests.mpim-bonn.mpg.de/martel/ | |
