Equivariant embeddings of differentiable spaces (Q2752985)

Continuous and differentiable actions by automorphisms of a compact Lie group \(G\) on a differentiable algebra \(A\) with compact spectrum, or respectively, on an affine differentiable space \(X\) are studied in this paper. In fact, the notion of differentiable action of \(G\) on \(X\) is introduced just here. NEWLINENEWLINENEWLINEIn the first case, the existence of a continuous linear representation \(G \to Gl(E)\), where \(E\) is the dual space of a finite-dimensional vector subspace \(F\) of \(A\), and also the existence of a \(G\)-equivariant epimorphism \(C^{\infty}\to E\) is proved. NEWLINENEWLINENEWLINEIn the second case, a differentiable linear representation and \(G\)-equivariant closed embedding theorems are obtained for compact differentiable spaces, which are closely related with the Spallek theory of differentiable spaces.