# invariant form (Lie algebras)

Let $V$ be a representation^{} of a Lie algebra^{} $\U0001d524$ over a field $k$. Then a bilinear form^{}
$B:V\times V\to k$ is invariant if

$$B(Xv,w)+B(v,Xw)=0.$$ |

for all $X\in \U0001d524,v,w\in V$. This criterion seems a little odd, but in the context of Lie algebras, it makes sense. For example, the map $\stackrel{~}{B}:V\to {V}^{*}$ given by $v\mapsto B(\cdot ,v)$ is equivariant if and only if $B$ is an invariant form.

Title | invariant form (Lie algebras) |
---|---|

Canonical name | InvariantFormLieAlgebras |

Date of creation | 2013-03-22 13:15:45 |

Last modified on | 2013-03-22 13:15:45 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 5 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 17B15 |