Finitely $C^\infty$-generated Hopf algebras

The definition of a finitely $C^\infty$-generated algebra is analogous to that of a holomorphically finitely generated algebra of Pirkovskii, in that it is defined as a quotient of an algebra of free $C^\infty$ functions of finite rank. One of the reasons for the interest in this new class of algebras is a good behaviour with respect to the projective tensor product, which makes it possible to study topological Hopf algebras with underlying algebras in this class. We discuss some general theory and also consider two non-trivial examples.