کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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4585173 | 1630525 | 2013 | 14 صفحه PDF | دانلود رایگان |
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).
Journal: Journal of Algebra - Volume 389, 1 September 2013, Pages 137–150