The authors introduce the basic theory of braid groups, highlighting several definitions showing their equivalence. This is followed by a treatment of the relationship between braids, knots and links. Important results then look at linearity and orderability.

link. C. Kassel, V. Turaev, Braid Groups, DOI: 10.1007/978-0-387-68548-9 2, c Springer Science+Business Media, LLC 2008 48 2 Braids, Knots, and Links Fig. 2.1. Knots and links in R3 Two geometric links L and L in M are isotopic if L can be deformed into L by an isotopy of M into itself. Here by an isotopy of M (into itself), we mean a continuous family of homeomorphisms {Fs : M → M }s∈I such that F0 = idM : M → M . The continuity of this family means that the mapping I → Top(M ), s → Fs is

introduce the braid groups and discuss some of their simple properties. 1.1.1 Basic deﬁnition We give an algebraic deﬁnition of the braid group Bn for any positive integer n. The deﬁnition is formulated in terms of a group presentation by generators and relations. Deﬁnition 1.1. The Artin braid group Bn is the group generated by n − 1 generators σ1 , σ2 , . . . , σn−1 and the “braid relations” σi σj = σj σi for all i, j = 1, 2, . . . , n − 1 with |i − j| ≥ 2, and σi σi+1 σi = σi+1 σi σi+1 for i =

i. This can be rewritten as Ui (Ui − (1 − t)In ) = tIn . Hence, Ui is invertible over Λ and its inverse is computed by ⎞ ⎛ 0 0 0 Ii−1 ⎜ 0 0 1 0 ⎟ ⎟. Ui−1 = t−1 (Ui − (1 − t)In ) = ⎜ ⎝ 0 0 ⎠ t−1 1 − t−1 0 0 0 In−i−1 The block form of the matrices U1 , . . . , Un−1 implies that Ui Uj = Uj Ui for all i, j with |i − j| ≥ 2. We also have Ui Ui+1 Ui = Ui+1 Ui Ui+1 for i = 1, . . . , n − 2. To check this, it is enough to verify the equality ⎛ 1−t ⎝ 1 0 t 0 0 ⎞⎛ 0 1 0⎠⎝0 1 0 ⎞⎛ ⎞ 0 0 1−t t 0 1 − t t

free left Hn−1 -module with basis {1, Tn−1 , Tn−1 Tn−2 , . . . , Tn−1 Tn−2 · · · T2 T1 } . Therefore, Hn ⊕ (Hn ⊗Hn−1 Hn ) is a free left Hn -module with basis {1} {1 ⊗ 1, 1 ⊗ Tn−1 , 1 ⊗ Tn−1 Tn−2 , . . . , 1 ⊗ Tn−1 Tn−2 · · · T2 T1 } . 4.2 The Iwahori–Hecke algebras 169 The map ϕ sends this basis to the set {1} {Tn , Tn Tn−1 , Tn Tn−1 Tn−2 , . . . , Tn Tn−1 Tn−2 · · · T2 T1 } , which by Proposition 4.21 is a basis of the left Hn -module Hn+1 . This implies that ϕ is an isomorphism.

λ∈Λ 4.5 Semisimple algebras and modules 183 Proof. Under the assumptions, A can be identiﬁed with the direct sum λ∈Λ Aλ . It follows from the deﬁnition of the product in A that each right multiplication Ra ∈ EndK (A), where a = (aλ )λ ∈ A, is the direct sum over λ ∈ Λ of the right multiplications Raλ . Therefore, the trace form , of A is the sum of the trace forms , λ of the algebras {Aλ }λ∈Λ , that is, (aλ )λ , (bλ )λ = aλ , b λ λ λ∈Λ for all (aλ )λ , (bλ )λ ∈ A. It follows that λ∈Λ J(Aλ