I am reading through Finite groups of Lie type_ conjugacy classes and complex characters by Roger Carter, and came across this passage where Carter is setting up a special class of module to give a representation $\rho$ of a lie algebra $\mathfrak{g}$:
Let $\frak{g}$ be a finite dimensional Lie algebra over an arbitrary field $k$, and let $M$ be a finite dimensional $k\frak{g}$-module.
Im assuming by this he means a $(\mathfrak{g},k)$-module because we are building the machinery to prove the Jacobson-Morozov theorem. For a $(\mathfrak{g},k)$-module though, $k$ has to be a maximal compact subgroup of the Lie group corresponding to $\frak{g}$ -- but arbitrary fields are not compact?
Was I wrong to assume that a $k\mathfrak{g}$-module was a $(\mathfrak{g},k)$-module by some different notation? Unfortunately, Carter doesn't elaborate much more on what he means by this. Any suggested readings would be appreciated, I am quite new to the study of lie algebras and don't really know what I am doing yet!