# Casimir operator

Let $\U0001d524$ be a semisimple Lie algebra^{}, and let $(\cdot ,\cdot )$ denote the Killing form^{}. If $\{{g}_{i}\}$ is a basis of $\U0001d524$, then there is a dual basis^{} $\{{g}^{i}\}$ with respect to the Killing form, i.e., $({g}_{i},{g}^{j})={\delta}_{ij}$.
Consider the element $\mathrm{\Omega}=\sum {g}_{i}{g}^{i}$ of the universal enveloping algebra of $\U0001d524$. This element, called the Casimir operator^{} is central^{} in the enveloping algebra, and thus commutes with the $\U0001d524$ action on any representation.

Title | Casimir operator |
---|---|

Canonical name | CasimirOperator |

Date of creation | 2013-03-22 13:52:53 |

Last modified on | 2013-03-22 13:52:53 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 5 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 17B20 |