Notation for modules. Given a module $A$ for a group $G$, and a subgroup $P\leq G$ with unipotent radical $U$, I have encountered the notation $[A,U]$ in a paper. Is this a standard module-theoretic notation, and if so, what does it mean.
In the specific case I am looking at, it works out that $[A,U]$ is equal to the submodule of the restriction of $A$ to $P$ generated by the fixed-point space of $A$ with respect to $U$, but whether this is the case in general I do not know. If anyone could enlighten me on this notation, it would be greatly appreaciated.
 A: In group theory it is standard to view $G$-modules $A$ as embedded in the semi-direct product $G \ltimes A$. Inside the semidirect product, the commutator subgroup $[A,U]$ makes sense for any subgroup $U \leq G$ and since $A$ is normal in $G \ltimes A$, we get $[A,U] \leq A$ and by the end we need make no reference to the semi-direct product. If we let $A$ be a right $G$-module written multiplicatively so that the $G$ action is written as exponentiation, then $$[A,U] = \langle [a,u] : a \in A, u \in U \rangle = \langle a^{-1} a^u : a \in A, u \in U \rangle$$
If you have a left $A$-module written additively and $G$-action written as multiplication, then we get $$[A,U] = \langle a - u\cdot a : a \in A, u \in U \rangle = \sum_{u \in U} \operatorname{im}(1-u)$$ which is just the sum of the images of $A$ under the nilpotent operators $1-u$, which is probably a fairly interesting thing to consider.
In some sense this is the dual of the centralizer: $A/[A,U]$ is the largest quotient of $A$ centralized by $U$.
