# Killing form

Let $\U0001d524$ be a finite dimensional Lie algebra^{} over a field $k$, and $X,Y\in \U0001d524$. Let ${\mathrm{ad}}_{X}:\U0001d524\to \U0001d524$ be the adjoint action,
${\mathrm{ad}}_{X}Y=[X,Y]$.

Then the Killing form on $\U0001d524$ is a bilinear map

$${B}_{\U0001d524}:\U0001d524\times \U0001d524\to k$$ |

given by

$${B}_{\U0001d524}(X,Y)=\mathrm{tr}({\mathrm{ad}}_{X}\circ {\mathrm{ad}}_{Y}).$$ |

The Killing form is invariant (http://planetmath.org/InvariantFormLieAlgebras) under the adjoint action and symmetric^{} (since trace is
symmetric).

Title | Killing form |
---|---|

Canonical name | KillingForm |

Date of creation | 2013-03-22 13:15:47 |

Last modified on | 2013-03-22 13:15:47 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 8 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 17B05 |