An inner automorphism is only an inner automorphism, but an inner endomorphism can be something strange. (Q433390)

The paper is based on a categorical description of inner automorphisms of groups and algebras over a field. For a group \(G\), let \(G\downarrow\mathbf{Grp}\) denote the comma category, consisting of morphisms \(G\to H\) in \(\mathbf{Grp}\) as objects and the obvious morphisms. Then inner automorphisms of \(G\) can be described as automorphisms of the forgetful functor \(G\downarrow\mathbf{Grp}\to\mathbf{Grp}\). A similar theorem holds for algebras over a field.NEWLINENEWLINE If automorphisms are replaced by endomorphisms, not much changes in the group case, while the algebra case is closely related to the classification of endomorphisms of the \(C^*\)-algebra of bounded operators on a Hilbert space. For arbitrary \(C^*\)-algebras, this structure is reflected by the Cuntz algebras. In the same vein, inner derivations are investigated for associative algebras and for Lie algebras.