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The reader should check by direct calculation that the pair (V, c) is a braided vector space.
Moreover, we have
(c − q id V ⊗V )(c + q −1 id V ⊗V ) = 0 .
For q = 1, one recovers example (i). For this reason, example (ii) is called a one-parameter
deformation of example (i).
1.2 Braid groups
Definition 1.2.1
Fix an integer n ≥ 3. The braid group B n on n strands is the group with n − 1 generators
σ 1 . . . σ n−1 and relations
σ i σ j = σ j σ i for |i − j| > 1.
σ i σ i+1 σ i = σ i+1 σ i σ i+1 for 1 ≤ i ≤ n − 1
We define for n = 2 the braid group B 2 as the free group with one generator and we let
B 0 = B 1 = {1} be the trivial group.
Remarks 1.2.2.
(i) The following pictures explain the name braid group:
σ i =
. . .
. . .
1 2 i i + 1
n
σ j σ i =
. . . . . . . . .
1 2 i i + 1 j j + 1 n
= σ i σ j
σ 1 σ 2 σ 1 = = = σ 2 σ 1 σ 2
(ii) There is a canonical surjection from the braid group to the symmetric group:
π : B n → S n
σ i ↦→ τ i,i+1 .
There is an important difference between the symmetric group S n and the braid group
B n : in the symmetric group S n the relation τ 2 i,i+1 = id holds. In contrast to the symmetric
group, the braid group is an infinite group without any non-trivial torsion elements, i.e.
without elements of finite order.
Let (V, c) be a braided vector space. For 1 ≤ i ≤ n − 1, define an automorphism of V ⊗n by
⎧
⎨ c ⊗ id V ⊗(n−2) for i = 1
c i := id
⎩ V ⊗(i−1) ⊗ c ⊗ id V ⊗(n−i−1) for 1 < i < n − 1
id V ⊗(n−2) ⊗ c for i = n − 1 .
We deduce from the axioms of a braided vector space that this defines a linear representation
of the braid group B n on the vector space V ⊗n :
2