It was J. Souriau [Structure des systèmes dynamiques. Maitrises de mathématique. Paris: Dunod (1970; Zbl 0186.58001)] who introduced the notion of a moment map for the first time. The tool deals with symmetries in symplectic or pre-symplectic geometry. The author of this paper, who was once a student of Souriau, is most enthusiastic in developing diffeology, the ideas of which date back to [J. M. Souriau, Lect. Notes Math. 836, 91–128 (1980; Zbl 0501.58010); K.-T. Chen, Bull. Am. Math. Soc. 83, 831–879 (1977; Zbl 0389.58001); Trans. Am. Math. Soc. 206, 83–98 (1975; Zbl 0301.58006); Ann. Math. (2) 97, 217–246 (1973; Zbl 0227.58003)]. He has already written a monograph on diffeology [Diffeology. Mathematical Surveys and Monographs 185. Providence, RI: American Mathematical Society (2013; Zbl 1269.53003)]. The principal objective in this memoir is to extend the notion of symplectic formalism and moment maps to diffeology. As the author has stated, “the moment map is just an object of the world of differential closed form, and there is no reason a priori that it could not be extended to diffeology which has a very well developed framework for De Rham’s calculus.”
Given a diffeological space \(X\) equipped with a closed \(2\)-form \(\omega\) and a diffeological group \(G\) acting on \(M\)with preserving \(\omega\), the space \(\mathcal{G}^{\ast}\) of left-invariant \(1\)-forms is called the space of momenta. The difficulty is that not every \(G\)-invariant closed form is exact, and that there is no reason to assume its primitives, if any, to be \(G\)-invariant. To overcome this difficulty, the author considers the space \(\mathrm{Paths}(X)\) of all smooth paths on \(X\), on which we get a differential \(1\)-form by integrating \(\omega\) along the paths. The moment map \(\Psi\) on \(\mathrm{Paths}(X)\) is then put down to the moment map \(\mu\) on \(X\) taking its values in some quotient of the space of moments in place of the space of moments itself. By taking \(G\) to be the group \(\mathrm{Diff}(X,\omega)\) of all diffeomorphisms of \(X\) preserving \(\omega\), we get a universal one. The last chapter is devoted to several examples involving diffeological groups or diffeological spaces essentially. The author’s approach reveals the fact that the theory of moment maps should proceed covariantly, avoiding such contravariant objects as Lie algebras or vector fields.