Braid group

review of symmetric groups

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

$B_n$
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 $\rho(\sigma_i)=\bar{R}_i$