A real representation of an algebra from a complex one? Here's a dumb question from a physicist. If I have a complex matrix irreducible representation of an algebra over the space $\mathbb{C}^n$, does that imply that this representation can be treated as a real (or pseudo-real) matrix representation of the same algebra over $\mathbb{R}^{2n}$? If yes, what are the conditions of irreducibility? If no, then what shall we generally obtain when considering the action of the algebra on these $2n$ real numbers? A simple example with smth like $\mathfrak{su}(2)$ would be appreciated. 
 A: A complex representation of a Lie algebra on $\mathbb C^n$ automatically is a real representation on $\mathbb R^{2n}$. Expressed in matrices, this corresponds to viewing a complex number $z=a+ib$ as the matrix $\begin{pmatrix}a & -b \\ b & a\end{pmatrix}$. If you apply this to the standard representation of $\mathfrak{su}(2)$ on $\mathbb C^2$ as $\begin{pmatrix}ix & -\bar z \\ z & -ix\end{pmatrix}$ then you get a representation as real $4\times 4$-matrizes of the form $\begin{pmatrix} 0 & -x & -a & -b \\x & 0 & b & -a\\ a & -b & 0 & x\\ b & a &-x & 0\end{pmatrix}$. 
Irreducibility is a slightly subtle question, since irreducibility of the complex representation on $\mathbb C^n$ means that there is no complex subspace that is invariant under the action, whereas on the real level, you are looking for invariant real subspaces in $\mathbb R^{2n}$. A simple example is given by $\mathfrak{sl}(2,\mathbb R)$. This can be viewed as acting on $\mathbb C^2$ by simply viewing real matrices as complex ones, and as a complex representation, this is irreducible. But of course $\mathbb R^2\subset\mathbb C^2$ (i.e. the points with both coordinates real) is invariant, so as a representation on $\mathbb R^4$, this is not irreducible. 
The general answer is that if you start with an irreducible complex representation, then it is also irreducible as a real representation if and only if it is not the complexification of a real representation. (This also is exactly the difference between the standard representations of $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb R)$ on $\mathbb C^2$.)
