# 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 IsotropyRepresentation 2013-03-22 12:42:28 2013-03-22 12:42:28 rmilson (146) rmilson (146) 6 rmilson (146) Definition msc 17B10 AdjointRepresentation