Braid group

review of symmetric groups

• 원소의 개수가 n인 집합의 전단사함수들의 모임
• \(n!\) 개의 원소가 존재함
• 대칭군의 부분군은 치환군(permutation group)이라 불림

 

 

presentation of symmetric groups

• 생성원 \(\sigma_1, \ldots, \sigma_{n-1}\)
  
• relations
  
  • \({\sigma_i}^2 = 1\)
    
  • \(\sigma_i\sigma_j = \sigma_j\sigma_i \mbox{ if } j \neq i\pm 1\)
    
  • \(\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}\\)
    

 

presentation of braid groups

\(B_n\)

generators \(\sigma_1,...,\sigma_{n-1}\)

relations (known as the braid or Artin relations)\[\sigma_i\sigma_j =\sigma_j \sigma_i\] whenever \(|i-j| \geq 2 \)

\(\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i \sigma_{i+1}\) for \(i = 1,..., n-2\)Yang-Baxter equation (YBE)