# quadratic Lie algebra

A Lie algebra^{} $\U0001d524$ is said to be *quadratic* if $\U0001d524$ as a representation^{} (under the adjoint action) admits a non-degenerate, invariant scalar product $(\cdot \mid \cdot )$ .

$\U0001d524$ being quadratic implies that the adjoint^{} and co-adjoint representations of $\U0001d524$ are isomorphic^{}.

Indeed, the non-degeneracy of $(\cdot \mid \cdot )$ implies that the induced map $\varphi :\U0001d524\to {\U0001d524}^{*}$ given by $\varphi (X)(Z)=(X\mid Z)$ is an isomorphism^{} of vector spaces^{}. Invariance of the scalar product^{} means that
$([X,Y]\mid Z)=-(Y\mid [X,Z])=(Y\mid [Z,X])$. This implies that $\varphi $ is a map of representations:

$$\varphi (a{d}_{X}(Y))(Z)=\varphi ([X,Y])(Z)=([X,Y]\mid Z)=(Y\mid [Z,X])=a{d}_{X}^{*}(\varphi (Y)(Z))$$ |

Title | quadratic Lie algebra |
---|---|

Canonical name | QuadraticLieAlgebra |

Date of creation | 2013-03-22 15:30:44 |

Last modified on | 2013-03-22 15:30:44 |

Owner | benjaminfjones (879) |

Last modified by | benjaminfjones (879) |

Numerical id | 6 |

Author | benjaminfjones (879) |

Entry type | Definition |

Classification | msc 17B10 |

Classification | msc 17B01 |

Related topic | quadraticAlgebra |

Defines | quadratic Lie algebra |