= Levi decomposition of a quiver automorphism group
{c}
For a finite-dimensional <quiver representation> $X$ over an <algebraically closed field> $k$, fix a <Krull-Schmidt decomposition> $X=\bigoplus_aM_a^{\oplus m_a}$ with pairwise nonisomorphic <indecomposable modules>. Acting by scalar <matrices> on the multiplicity spaces embeds $L=\prod_a\operatorname{GL}_{m_a}(k)$ into $\operatorname{Aut}_Q(X)$. It splits reduction modulo the <Jacobson radical> $J=J(\operatorname{End}_Q(X))$, giving
$$
\operatorname{Aut}_Q(X)=(1+J)\rtimes L.
$$
The <subgroup> $1+J$ is closed, normal and <unipotent>, because $J$ is a <nilpotent ideal>. This explicit section works in every <characteristic>. The <algebraically closed field> assumption ensures that the residue <division rings> of the indecomposable <endomorphism rings> are $k$; without it this particular product of split <general linear groups> need not be the quotient.
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