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Artin group

In mathematics, an Artin group (or generalized braid group) is a group with a presentation of the form

$\left\langle x_1,x_2,\ldots,x_n|\langle x_1, x_2 \rangle^{m_{1,2}}=\langle x_2, x_1 \rangle^{m_{1,2}}, \langle x_1, x_3 \rangle^{m_{1,3}}=\langle x_3, x_1 \rangle^{m_{1,3}}, \ldots \langle x_{n-1}, x_n \rangle^{m_{n-1,n}}=\langle x_{n-1}, x_n \rangle^{m_{n,n-1}}\right\rangle$

where

$m_{i,j} \in {0,1,\ldots, \infty}$ .

For $m < \infty$ , $\langle x_i, x_j \rangle^m$ represents an alternating product of $x i$ and $x j$ of length $m$ , beginning with $x i$ . (For example, $\langle x_i, x_j \rangle^3 = x_ix_jx_i$ and $\langle x_i, x_j \rangle^4 = x_ix_jx_ix_j$ .) If $m=\infty$ , then there is no relation for $x i$ and $x j$ .

The $m i, j$ 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 ${x_i}^2=1$ .

Braid groups are examples of Artin groups, with Coxeter matrix $m i, i + 1 = 3$ and $m i, j = 2$ for $| i - j | > 1.$

Categories: Group theory

Last updated: 05-21-2005 18:56:22

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10-26-2009 08:16:03
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Artin group