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Lie theory and coverings of finite groups

Research output: Contribution to journalArticlepeer-review

S. Majid, K. Rietsch

Original languageEnglish
Pages (from-to)137-150
Number of pages14
JournalJournal of Algebra
Volume389
Issue numberN/A
DOIs
E-pub ahead of print1 Sep 2013

Documents

  • 1209.0045v2.pdf

    1209.0045v2.pdf, 482 KB, application/pdf

    Uploaded date:21 Jul 2015

    Version:Submitted manuscript

King's Authors

Abstract

We introduce the notion of an ‘inverse property’ (IP) quandle C which we propose as the right notion of ‘Lie algebra’ in the category of sets. For any IP-quandle we construct an associated group GC. For a class of IP-quandles which we call ‘locally skew’, and when GC is finite, we show that the noncommutative de Rham cohomology H1(GC) is trivial aside from a single generator θ that has no classical analogue. If we start with a group G then any subset C⊆G∖{e} which is ad-stable and inversion-stable naturally has the structure of an IP-quandle. If C also generates G then we show that GC↠G with central kernel, in analogy with the similar result for the simply-connected covering group of a Lie group. We prove that this ‘covering map’ GC↠G is an isomorphism for all finite crystallographic reflection groups W with C the set of reflections, and that C is locally skew precisely in the simply laced case. This implies that H1(W)=k when W is simply laced, proving in particular a conjecture for Sn in Majid (2004) [12]. We also consider C=ZP1∪ZP1 as a locally skew IP-quandle ‘Lie algebra’ of SL2(Z) and show that GC≅B3, the braid group on 3 strands. The map B3↠SL2(Z) which therefore arises naturally as a covering map in our theory, coincides with the restriction of the usual universal covering map to the inverse image of SL2(Z).

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