# root system

A root system^{} is a key notion in the classification and the
representation theory of reflection groups and of semi-simple Lie
algebras. Let $E$ be a Euclidean vector space with inner product
$(\cdot ,\cdot )$. A root system is a finite spanning set $R\subset E$
such that for every $u\in R$, the orthogonal^{} reflection^{}

$$v\mapsto v-2\frac{(u,v)}{(u,u)}u,v\in E$$ |

preserves $R$.

A root system is called *crystallographic* if
$2\frac{(u,v)}{(u,u)}$ is an integer for all $u,v\in R$.

A root system is called reduced if for all $u\in R$, we have $ku\in R$ for $k=\pm 1$ only.

We call a root system indecomposable if there is no proper decomposition $R={R}^{\prime}\cup {R}^{\prime \prime}$ such that every vector in ${R}^{\prime}$ is orthogonal to every vector in ${R}^{\prime \prime}$.

Title | root system |

Canonical name | RootSystem |

Date of creation | 2013-03-22 13:11:30 |

Last modified on | 2013-03-22 13:11:30 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 13 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 17B20 |

Related topic | SimpleAndSemiSimpleLieAlgebras2 |

Related topic | LieAlgebra |

Defines | reduced root system |

Defines | root |

Defines | root space |

Defines | root decomposition |

Defines | indecomposable |

Defines | reduced |

Defines | crystallographic |