Actions of Braid Groups
In analogy with the action of the symmetric group by permutations, in various mathematical settings there exists a natural action the braid group on n-tuples of objects or on the n-folded tensor product that involves some "twists". Consider an arbitrary group G and let X be the set of all n-tuples of elements of G whose product is the identity element of G. Then Bn acts on X in the following fashion:
Thus the elements xi and xi+1 exchange places and, in addition, xi is twisted by the inner automorphism corresponding to xi+1 — this ensures that the product of the components of x remains the identity element. It may be checked that the braid group relations are satisfied and this formula indeed defines a group action of Bn on X. As another example, a braided monoidal category is a monoidal category with a braid group action. Such structures play an important role in modern mathematical physics and lead to quantum knot invariants.
Read more about this topic: Braid Group
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