2012/09/01 by Majid, Shahn, Rietsch, Konstanze · 1 citation
#FOS: Mathematics #Group Theory (math.GR) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.1209.0045
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 \twoheadrightarrow G with central kernel, in analogy with the similar result for the simply-connected covering group of a Lie group. We prove that GC\twoheadrightarrow 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\twoheadrightarrow SL2(Z) which arises naturally as a covering map in our theory, coincides with the restriction of the universal covering map \widetilde SL2(R)→ SL2(R) to the inverse image of SL2(Z).