Braid group

review of symmetric groups

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

presentation of symmetric groups

• <math>S_n</math>
• generators <math>\sigma_1, \ldots, \sigma_{n-1}</math>
• relations
  • <math>{\sigma_i}^2 = 1</math>
  • <math>\sigma_i\sigma_j = \sigma_j\sigma_i \mbox{ if } j \neq i\pm 1</math>
  • <math>\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}</math>

presentation of braid groups

• <math>B_n</math>
• generators <math>\sigma_1,...,\sigma_{n-1}</math>
• relations (known as the braid or Artin relations):
  • <math>\sigma_i\sigma_j =\sigma_j \sigma_i</math> whenever <math>|i-j| \geq 2 </math>
  • <math>\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i \sigma_{i+1}</math> for <math>i = 1,..., n-2</math>
• Yang-Baxter equation (YBE)
• For a solution of the YBE <math>\bar{R}</math>, we can construct a representation <math>\rho</math> of the braid group by

<math>

\rho : B_n \to \rm{Aut}(V^{\otimes n}) </math> where <math>\rho(\sigma_i)=\bar{R}_i</math>

There is also a natural surjective morphism from <math>B_n</math> to the symmetric group <math>\mathfrak{S}_n</math>, given on the generators by <math>B_n\ni\sigma_i\mapsto s_i\in \mathfrak{S}_n</math>, <math>i=1,\dots,n-1</math>. For a braid <math>\beta\in B_n</math>, we denote <math>p_{\beta}</math> its image in <math>\mathfrak{S}_n</math>, and refer to <math>p_{\beta}</math> as to the underlying permutation of <math>\beta</math>.

examples

• in a braid diagram, read from bottom to top and we number all strands of the braid with the indices it starts at the bottom

• read the braid word from left to right accordingly.
• For instance, the braid word corresponding to the braid above is <math>\sigma_1^{-1}\sigma_2\sigma_1^{-1}\sigma_2\sigma_1^{-1}</math>

Markov moves

• braid group version of Reidemeister moves

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxZ3NjMGpGUWI0QkE/edit

related items

• Jones polynomials
• Hecke algebra
• Loop braid group

encyclopedia

• http://en.wikipedia.org/wiki/Braid_group

expositions

• Abad, Camilo Arias. 2014. “Introduction to Representations of Braid Groups.” arXiv:1404.0724 [math], April. http://arxiv.org/abs/1404.0724.

메타데이터

위키데이터

• ID : Q220409

Spacy 패턴 목록

• [{'LOWER': 'braid'}, {'LEMMA': 'group'}]