Lie theory and coverings of finite groups.

Abstract: We introduce the notion of an ‘inverse property’ (IP) quandle which we propose as the right notion of ‘Lie algebra’ in the category of sets. For any IP-quandle we construct an associated group . For a class of IP-quandles which we call ‘locally skew’, and when is finite, we show that the noncommutative de Rham cohomology is trivial aside from a single generator θ that has no classical analogue. If we start with a group G then any subset which is ad-stable and inversion-stable naturally has the structure of an IP-quandle. If also generates G then we show that 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’ is an isomorphism for all finite crystallographic reflection groups W with the set of reflections, and that is locally skew precisely in the simply laced case. This implies that when W is simply laced, proving in particular a conjecture for in Majid (2004) [12]. We also consider as a locally skew IP-quandle ‘Lie algebra’ of and show that , the braid group on 3 strands. The map 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 . [Copyright &y& Elsevier]