# simple and semi-simple Lie algebras

A Lie algebra^{} is called simple if it has no proper ideals^{} and is not abelian^{}. A Lie algebra
is called semi-simple^{} if it has no proper solvable^{} ideals and is not abelian.

Let $k=\mathbb{R}$ or $\u2102$. Examples of simple algebras are $\U0001d530{\U0001d529}_{n}k$, the Lie algebra
of the special linear group^{} (traceless matrices), $\U0001d530{\U0001d52c}_{n}k$, the Lie algebra of the special
orthogonal group^{} (skew-symmetric matrices), and $\U0001d530{\U0001d52d}_{2n}k$ the Lie algebra of the symplectic group. Over $\mathbb{R}$, there are other simple Lie algebas, such as $\U0001d530{\U0001d532}_{n}$, the Lie algebra of the special unitary group
(skew-Hermitian matrices). Any
semi-simple Lie algebra is a direct product^{} of simple Lie algebras.

Simple and semi-simple Lie algebras are one of the most widely studied classes of algebras for a number of reasons. First of all, many of the most interesting Lie groups have semi-simple Lie algebras. Secondly, their representation theory is very well understood. Finally, there is a beautiful classification of simple Lie algebras.

Over $\u2102$, there are 3 infinite series of simple Lie algebras: $\U0001d530{\U0001d529}_{n}$, $\U0001d530{\U0001d52c}_{n}$ and $\U0001d530{\U0001d52d}_{2n}$, and 5 exceptional simple Lie algebras ${\U0001d524}_{2},{\U0001d523}_{4},{\U0001d522}_{6},{\U0001d522}_{7}$, and ${\U0001d522}_{8}$. Over $\mathbb{R}$ the picture is more complicated, as several different Lie algebras can have the same complexification (for example, $\U0001d530{\U0001d532}_{n}$ and $\U0001d530{\U0001d529}_{n}\mathbb{R}$ both have complexification $\U0001d530{\U0001d529}_{n}\u2102$).

Title | simple and semi-simple Lie algebras |

Canonical name | SimpleAndSemisimpleLieAlgebras |

Date of creation | 2013-03-22 13:11:28 |

Last modified on | 2013-03-22 13:11:28 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 9 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 17B20 |

Related topic | LieAlgebra |

Related topic | LieGroup |

Related topic | RootSystem |

Related topic | RootSystemUnderlyingASemiSimpleLieAlgebra |

Defines | simple Lie algebra |

Defines | semi-simple Lie algebra |

Defines | semisimple Lie algebra |

Defines | simple |

Defines | semi-simple |

Defines | semisimple^{} |