# Coset Space as a Representation of a Lie Algebra

I'm reading through some notes (about the use of Lie groups/algebras in physics) obtained from a friend from a class that took a while back, and I can't quite figure out where one thing regarding some cosets spaces came from.

The setup is as follows:

Consider the group $ISO(m,n)$ of isometries of $\mathbb{R}^{m,n}$, i.e. maps $\varphi:\mathbb{R}^{m+n}\rightarrow\mathbb{R}^{m+n}$ that preserve $-x_1^2+\cdots-x_m^2+x_{m+1}^2+\cdots+x_{m+n}^2$.

The Lie algebra $\mathfrak{iso}(m,n)$ of this group can be written in terms of generators $J_{\mu\nu}$ and $K_{\mu}$ (where $0\leq\mu<\nu< m+n$) satisfying:

$$[J_{\mu\nu},J_{\rho\sigma}]=i\hbar(\eta_{\mu\sigma}J_{\nu\rho}-\eta_{\mu\rho}J_{\nu\sigma}+\eta_{\nu\sigma}J_{\mu\rho}-\eta_{\nu\rho}J_{\mu\sigma}),$$

$$[K_{\mu},J_{\nu\rho}]=i\hbar(\eta_{\mu\rho}K_{\nu}-\eta_{\mu\nu}K_{\rho}), \qquad [K_{\mu},K_{\nu}]=0.$$

Then the $J_{\mu\nu}$'s can be thought of as the infinitesimal generators for the subgroup $SO(m,n)<ISO(m,n)$. We write an arbitrary element of the Lie algebra as

$$X=-\frac{i}{\hbar}\left(\frac{1}{2}\omega^{\mu\nu}J_{\mu\nu}+c^{\mu}K_{\mu}\right).$$

(The factors of $\hbar$, etc., aren't terribly important here -- they are simply to get the units correct in whatever physical story one wants to tell using this language.)

Restricting to the case of $ISO(1,3)$ relevant to special relativity and such, the claim is that the coset space $M=ISO(1,3)/SO(1,3)$ is a pseudo-Euclidean space with metric $\eta_{\mu\nu}$ (i.e. Minkowski space), and moreover that this coset space can be written as a representation $\rho$ of the algebra $\mathfrak{iso}(1,3)$ in the following way:

$$\rho(X)= \left( \begin{array}{cc} \omega^{\mu}_{\nu} & c^{\mu} \\ 0 & 0 \\ \end{array} \right),$$

(where $\omega^{\mu\nu}=-\omega^{\nu\mu}$) or in other words, $X$ represents an infinitesimal transformation on $M$ that does the following:

$$\left( \begin{array}{c} dx^{\mu} \\ 0 \\ \end{array} \right)=\left( \begin{array}{cc} \omega^{\mu}_{\nu} & c^{\mu} \\ 0 & 0 \\ \end{array} \right)\left( \begin{array}{c} x^{\mu} \\ 1 \\ \end{array} \right).$$

I have absolutely no clue (and there is zero explanation in the notes as to) how one obtains these matrices. Any help to this end would be tremendously appreciated.

It turns out that this is fairly easy, so I will simply record the answer here in case it is useful for anyone else later.

All that is going on here is the natural action of the Lie algebra $\mathfrak{g}$ on itself (given by the Lie bracket) being projected onto the coset space. More specifically, writing a general element of $\mathfrak{iso}(1,3)$ as

$$X=-\frac{i}{\hbar}\left(\frac{1}{2}\omega^{\mu\nu}J_{\mu\nu}+c^{\mu}K_{\mu}\right),$$

and an element of $\mathfrak{iso}(1,3)/\mathfrak{so}(1,3)$ as

$$Y=-\frac{i}{\hbar}x^{\mu}K_{\mu},$$

Then a quick computation yields

$$[X,Y]\propto (\omega_{\mu}^{\nu}x^{\mu}+c^{\mu})K_{\mu} + (\text{some stuff})J_{\mu\nu}.$$

The matrices given in the question then amount to writing the parameter space of the coset space as $(x^{\mu},1)$ (where here the 1 just represents the $\mathfrak{so}(1,3)$ stuff that has been collapsed together in the quotient), and so we can write the above commutator in matrix form as

$$[Y,X]=\left( \begin{array}{cc} \omega^{\nu}_{\mu} & c^{\mu} \\ 0 & 0 \\ \end{array} \right)\left( \begin{array}{c} x^{\mu} \\ 1 \\ \end{array} \right),$$

which, of course, gives us a representation of $\mathfrak{iso}(1,3)$ on the parameter space of $\mathfrak{iso}(1,3)/\mathfrak{so}(1,3)$ as

$$\rho(X)=\left( \begin{array}{cc} \omega^{\nu}_{\mu} & c^{\mu} \\ 0 & 0 \\ \end{array} \right).$$

If anyone is ever reading this and needs some clarification, just let me know!