# isotropy representation

Let $\U0001d524$ be a Lie algebra^{}, and $\U0001d525\subset \U0001d524$ a subalgebra^{}. The
isotropy representation of $\U0001d525$ relative to $\U0001d524$ is the naturally
defined action of $\U0001d525$ on the quotient vector space $\U0001d524/\U0001d525$.

Here is a synopsis of the technical details. As is customary, we will use

$$b+\U0001d525,b\in \U0001d524$$ |

to denote the coset elements of $\U0001d524/\U0001d525$.
Let $a\in \U0001d525$ be given. Since $\U0001d525$ is invariant with respect to
${ad}_{\U0001d524}(a)$, the adjoint action factors through the quotient to
give a well defined endomorphism^{} of $\U0001d524/\U0001d525$. The action is given
by

$$b+\U0001d525\mapsto [a,b]+\U0001d525,b\in \U0001d524.$$ |

This is the action alluded to in the first paragraph.

Title | isotropy representation |
---|---|

Canonical name | IsotropyRepresentation |

Date of creation | 2013-03-22 12:42:28 |

Last modified on | 2013-03-22 12:42:28 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 6 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 17B10 |

Related topic | AdjointRepresentation |