# classification of finite-dimensional representations of semi-simple Lie algebras

If $\U0001d524$ is a semi-simple Lie algebra, then we say that an representation^{} $V$
has highest weight $\lambda $, if there is a vector $v\in {V}_{\lambda}$, the weight space of
$\lambda $, such that $Xv=0$ for $X$ in any positive root space, and $v$ is called a highest
vector, or vector of highest weight.

There is a unique (up to isomorphism^{}) irreducible
finite dimensional representation of $\U0001d524$ with highest weight $\lambda $ for any dominant
weight $\lambda \in {\mathrm{\Lambda}}_{W}$, where ${\mathrm{\Lambda}}_{W}$ is the weight lattice^{} of $\U0001d524$, and
every irreducible representation of $\U0001d524$ is of this type.

Title | classification of finite-dimensional representations of semi-simple Lie algebras |
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Canonical name | ClassificationOfFinitedimensionalRepresentationsOfSemisimpleLieAlgebras |

Date of creation | 2013-03-22 13:11:40 |

Last modified on | 2013-03-22 13:11:40 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 5 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 17B20 |

Defines | highest weight |

Defines | highest vector |

Defines | vector of highest weight |

Defines | highest weight representation |