Braid Group - Actions of Braid Groups

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 B_n acts on X in the following fashion:

$\sigma_i(x_1,\ldots,x_{i-1},x_i, x_{i+1},\ldots, x_n)= (x_1,\ldots, x_{i-1}, x_{i+1}, x_{i+1}^{-1}x_i x_{i+1}, x_{i+2},\ldots,x_n).$

Thus the elements x_i and x_i+1 exchange places and, in addition, x_i is twisted by the inner automorphism corresponding to x_i+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 B_n 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.

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