On the group of automorphisms of an analytic group (Q2777718)

Let \(G\) be an analytic group and \(L(G)\) be the Lie algebra of \(G\). In the paper various conditions are studied under which the group \(\text{Aut }G\) of all topological group automorphisms of \(G\) is almost algebraic, i.e., when \(\text{Aut }G\) is a subgroup of finite index in an algebraic subgroup of \(GL(L(G)).\) It is proved, for example, that \(\text{Aut }G\) is almost algebraic if and only if the restriction map \(\tau\rightarrow \tau\mid T_{1}\) of \(\text{Aut}_{T}G\rightarrow \text{Aut } T_{1}\) has finite image, where \(T\) is a maximal torus of \(G,\text{Aut}_{T}G=\{\tau\in \text{Aut } G: \tau(T)=T\}\), and \(T_{1}=T\bigcap R\), \(R\) being the radical of \(G.\) It is proved as a corollary that if \(G\) is an analytic group for which the radical has a maximal torus of dimension at most 1, then \(\text{Aut}G\) is an almost algebraic group. For every natural number \(n\) an example is constructed of an analytic group whose maximal central torus is of dimension \(n\) and whose automorphism group is almost algebraic.