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Lie theory and coverings of finite groups
Quantum Algebra · Submitted 1 Sep 2012
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim If C also generates G then we show that G_C \twoheadrightarrow G with central kernel, in analogy with the similar result for the simply-connected covering group of a Lie group.
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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 G_C. For a class of IP quandles which we call `locally skew' and when G_C is finite we show that the noncommutative de Rham cohomology H^1(G_C) is trivial aside from a single generator \theta that has no classical analogue. If we start with a group G then any subset C\subseteq G\setminus {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 G_C \twoheadrightarrow G with central kernel, in analogy with the similar result for the simply-connected covering group of a Lie group. We prove that G_C\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 H^1(W)=k when W is simply laced, proving in particular a previous conjecture for S_n. We obtain similar results for the dihedral groups D_{6m}. We also consider C=Z P^1\cup Z P^1 as a locally skew IP-quandle `Lie algebra' of SL_2(Z) and show that G_C\cong B_3, the braid group on 3 strands. The map B_3\twoheadrightarrow SL_2(Z) which arises naturally as a covering map in our theory, coincides with the restriction of the universal covering map \widetilde {SL_2(R)}\to SL_2(R) to the inverse image of SL_2(Z).
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 1.00 | 0.93 | 0.09 | hedged possibility 0.53 |
| s1 | To any IP quandle we construct an associated group G_C. | contribution 0.87 | 0.94 | 0.06 | asserted 1.00 |
| s2 | For a class of IP quandles which we call `locally skew' and when G_C is finite we show that the noncommutative de Rham cohomology H^1(G_C) is trivial aside from a single generator \theta that has no classical analogue. | result 1.00 | 0.94 | 0.20 | asserted 1.00 |
| s3 | If we start with a group G then any subset C\subseteq G\setminus {e} which is ad-stable and inversion-stable naturally has the structure of an IP quandle. | contribution 0.51 | 0.75 | 0.04 | asserted 0.99 |
| s4 | If C also generates G then we show that G_C \twoheadrightarrow G with central kernel, in analogy with the similar result for the simply-connected covering group of a Lie group. | result 0.94 | 0.94 | 0.17 | asserted 1.00 |
| s5 | We prove that G_C\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. | result 0.99 | 0.95 | 0.24 | asserted 1.00 |
| s6 | This implies that H^1(W)=k when W is simply laced, proving in particular a previous conjecture for S_n. | interpretation 0.61 | 0.85 | 0.27 | asserted 1.00 |
| s7 | We obtain similar results for the dihedral groups D_{6m}. | result 0.89 | 0.94 | 0.06 | asserted 0.99 |
| s8 | We also consider C=Z P^1\cup Z P^1 as a locally skew IP-quandle `Lie algebra' of SL_2(Z) and show that G_C\cong B_3, the braid group on 3 strands. | result 0.88 | 0.95 | 0.07 | asserted 1.00 |
| s9 | The map B_3\twoheadrightarrow SL_2(Z) which arises naturally as a covering map in our theory, coincides with the restriction of the universal covering map \widetilde {SL_2(R)}\to SL_2(R) to the inverse image of SL_2(Z). | result 0.81 | 0.79 | 0.12 | asserted 1.00 |
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Original abstract
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 G_C. For a class of IP quandles which we call `locally skew' and when G_C is finite we show that the noncommutative de Rham cohomology H^1(G_C) is trivial aside from a single generator \theta that has no classical analogue. If we start with a group G then any subset C\subseteq G\setminus {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 G_C \twoheadrightarrow G with central kernel, in analogy with the similar result for the simply-connected covering group of a Lie group. We prove that G_C\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 H^1(W)=k when W is simply laced, proving in particular a previous conjecture for S_n. We obtain similar results for the dihedral groups D_{6m}. We also consider C=Z P^1\cup Z P^1 as a locally skew IP-quandle `Lie algebra' of SL_2(Z) and show that G_C\cong B_3, the braid group on 3 strands. The map B_3\twoheadrightarrow SL_2(Z) which arises naturally as a covering map in our theory, coincides with the restriction of the universal covering map \widetilde {SL_2(R)}\to SL_2(R) to the inverse image of SL_2(Z).
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