Braid group
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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
- \(S_n\)
- generators \(\sigma_1, \ldots, \sigma_{n-1}\)
- 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)
- For a solution of the YBE \(\bar{R}\), we can construct a representation \(\rho\) of the braid group by
\[ \rho : B_n \to \rm{Aut}(V^{\otimes n}) \] where \(\rho(\sigma_i)=\bar{R}_i\)
There is also a natural surjective morphism from \(B_n\) to the symmetric group \(\mathfrak{S}_n\), given on the generators by \(B_n\ni\sigma_i\mapsto s_i\in \mathfrak{S}_n\), \(i=1,\dots,n-1\). For a braid \(\beta\in B_n\), we denote \(p_{\beta}\) its image in \(\mathfrak{S}_n\), and refer to \(p_{\beta}\) as to the underlying permutation of \(\beta\).
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 \(\sigma_1^{-1}\sigma_2\sigma_1^{-1}\sigma_2\sigma_1^{-1}\)
Markov moves
- braid group version of Reidemeister moves
computational resource
encyclopedia
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'}]