isotropy representation

Let 𝔤 be a Lie algebraMathworldPlanetmath, and 𝔥𝔤 a subalgebraPlanetmathPlanetmathPlanetmath. The isotropy representation of 𝔥 relative to 𝔤 is the naturally defined action of 𝔥 on the quotient vector space 𝔤/𝔥.

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


to denote the coset elements of 𝔤/𝔥. Let a𝔥 be given. Since 𝔥 is invariant with respect to ad𝔤(a), the adjoint action factors through the quotient to give a well defined endomorphismMathworldPlanetmathPlanetmathPlanetmath of 𝔤/𝔥. The action is given by


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