Do you know of any literature/references for `braidifying' a group? That is, we can think of the symmetric group acting on n elements, and that we then attach a braiding group B_n by `remembering' how we permuted the elements. I'm interested in if something similar can be done for other permutation groups. That is, given a G that acts faithfully on a set X = {1, 2, ... n}, can we find a braiding group B(G) that keeps track how the permutations were performed? The problem is that there is a nice `canonical' way of choosing the generators for the symmetric group that allows you to make sense of the corresponding braiding. But, if I have a generator that acts on a set as, say (154)(23), should the braid corresponding to transposing 2 and 3 be behind or in front of those that cycle 1, 5, and 4? This freedom of choice makes the construction not well-defined, but I'm hoping that the construction is still well-defined up to some condition. One way of approaching this would be to think of the algebraic conditions the ordinary braiding group B_n satisfies. That is, B(G) should be torsion-free, and there exists a surjective hom from B(G) to G by forgetting the braids themselves. This together with some other conditions might specify B(G) (almost) uniquely.
@VisualMath
3 жыл бұрын
Interesting question. To the best of my knowledge, what comes close to what you are asking for is studied (quite a lot actually) under the term "Artin-Tits group": en.wikipedia.org/wiki/Artin%E2%80%93Tits_group giving credits to Tits who came up with this idea. This is huge class of groups, basically one for each labeled graph, and they satisfy properties very similar to the classical braid group. What makes them tempting is that they include "braidizations" of many groups appearing in mathematics, most prominently Weyl groups. They also appear in geometry, see e.g. wbln.centre-mersenne.org/item/10.5802/wbln.17.pdf which gives a pretty nice overview. For low-dimensional topology and braid pictures see arxiv.org/abs/math/9907194 For general groups I am skeptical to have nicely behaved "braidizations" - groups are wild beasts ;-)
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