Lie theory and coverings of finite groups

الأصل · EN

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. To 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 θthat 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 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 previous conjecture for Sₙ. We obtain similar results for the dihedral groups D₆ₘ. We also consider C=Z P¹∪ Z P¹ 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 arises naturally as a covering map in our theory, coincides with the restriction of the universal covering map SL₂(R)→ SL₂(R) to the inverse image of SL₂(Z).