isotropy representation

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

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

b+\mathfrak{h},\,b\in\mathfrak{g}

to denote the coset elements of $\mathfrak{g}/\mathfrak{h}$ . Let $a\in\mathfrak{h}$ be given. Since $\mathfrak{h}$ is invariant with respect to $\mathop{\mathrm{ad}}\nolimits_{\mathfrak{g}}(a)$ , the adjoint action factors through the quotient to give a well defined endomorphism of $\mathfrak{g}/\mathfrak{h}$ . The action is given by

b+\mathfrak{h}\mapsto[a,b]+\mathfrak{h},\quad b\in\mathfrak{g}.

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