Artin group
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In mathematics, an Artin group (or generalized braid group) is a group with a presentation of the form
- <math>\left\langle x_1,x_2,\ldots,x_n|\langle x_1, x_2 \rangle^{m_{1,2}}=\langle x_2, x_1 \rangle^{m_{1,2}}, \langle x_1, x_3 \rangle^{m_{1,3}}=\langle x_3, x_1 \rangle^{m_{1,3}}, \ldots \langle x_{n-1}, x_n \rangle^{m_{n-1,n}}=\langle x_{n-1}, x_n \rangle^{m_{n,n-1}}\right\rangle</math>
where
- <math>m_{i,j} \in {2,3,\ldots, \infty}</math>.
For <math>m < \infty</math>, <math>\langle x_i, x_j \rangle^m</math> represents an alternating product of <math>x_i</math> and <math>x_j</math> of length <math>m</math>, beginning with <math>x_i</math>. (For example, <math>\langle x_i, x_j \rangle^3 = x_ix_jx_i</math> and <math>\langle x_i, x_j \rangle^4 = x_ix_jx_ix_j</math>.) If <math>m=\infty</math>, then there is no relation for <math>x_i</math> and <math>x_j</math>.
The <math>m_{i,j}</math> can be organized into a symmetric matrix, known as the Coxeter matrix of the group. Each Artin group has as a quotient the Coxeter group with the same set of generators and Coxeter matrix. The kernel of the homomorphism to the associated Coxeter group is generated by relations of the form <math>{x_i}^2=1</math>.
Braid groups are examples of Artin groups, with Coxeter matrix <math>m_{i,i+1} = 3</math> and <math>m_{i,j}=2</math> for <math>|i-j|>1.</math>
