Lie theory and coverings of finite groups

We introduce the notion of an ‘inverse property’ (IP) quandle CC which we propose as the right notion of ‘Lie algebra’ in the category of sets. For any IP-quandle we construct an associated group GCGC. For a class of IP-quandles which we call ‘locally skew’, and when GCGC is finite, we show that the noncommutative de Rham cohomology H1(GC)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}C⊆G∖{e} which is ad-stable and inversion-stable naturally has the structure of an IP-quandle. If CC also generates G   then we show that GC↠GGC↠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↠GGC↠G is an isomorphism for all finite crystallographic reflection groups W   with CC the set of reflections, and that CC is locally skew precisely in the simply laced case. This implies that H1(W)=kH1(W)=k when W   is simply laced, proving in particular a conjecture for SnSn in Majid (2004) [12]. We also consider C=ZP1∪ZP1C=ZP1∪ZP1 as a locally skew IP-quandle ‘Lie algebra’ of SL2(Z)SL2(Z) and show that GC≅B3GC≅B3, the braid group on 3 strands. The map B3↠SL2(Z)B3↠SL2(Z) which therefore arises naturally as a covering map in our theory, coincides with the restriction of the usual universal covering map SL2(R)˜→SL2(R) to the inverse image of SL2(Z)SL2(Z).