Closed subgroup theorem

Abstract: © Andrée C. Ehresmann et les auteurs, 1989, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Abstract: Let M be a (C, Hausdorff, paracompact) manifold, and let R be an equivalence relation on M. Then R is called regular if the quotient M/R is a (not necessarily Hausdorff) manifold in such a way that the canonical projection TTR : M-+M/R is a submersion. For results on regular relations cf. Palais [1], Serre [2]. The following characterization of regularity is wellknown (cf. Serre [2, LG, Chapter 3, §12]): R is regular if and only if it is a submanifold (with the subspace topology) of M X Afin such a way that the map (m, m')-+m from R onto M is a submersion. The purpose of this note is to announce a different characterization of regularity. Proofs will appear elsewhere (Sussmann [3]). Our condition is motivated in a natural way by Systems Theory. As will be shown in [4], Theorem 2 is precisely what is needed to show that, under fairly general conditions, every finite-dimensional \"controllable\" nonlinear system has a realization which is both \"controllable\" and observable. Here we shall not pursue this line. Rather, we shall state our condition and show that it is a rather natural generalization of the closed subgroup theorem. Let X be a vector field on an open subset of M. We say that X is a symmetry vector field of R if, whenever (m, m') e R9 it follows that (Xt(m), Xt{m')) e R for every real t for which Xt(m) and Xt(m) are both defined (here t-+Xt(m) is the integral curve of X which passes through m when t=0). Let S°°CR, M) denote the set of all C vector fields X defined on open subsets of M that are symmetry vector fields for R. It is not difficult to show that S°°(R, M) is a presheaf of Lie algebras of vector fields. If L is a set of vector fields defined on open subsets of M, we say that L is transitive if, for every me M, the vectors X(m), X e L, span the tangent space of M at m. If A is a subset ofAfxM, we call L A-transitive if, for every (m, m') e A, the tangent space of M at m is spanned by the

Abstract: Given an axiomatic account of the category of locales the closed subgroup theorem is proved. The theorem is seen as a consequence of a categorical account of the Hofmann-Mislove theorem. The categorical account has an order dual providing a new result for locale theory: every compact subgroup is necessarily fitted.

Abstract: In this chapter we introduce the exponential map of a Lie group, which is a canonical smooth map from the Lie algebra into the group, mapping lines through the origin in the Lie algebra to one-parameter subgroups. As our first application, we prove the closed subgroup theorem, which says that every topologically closed subgroup of a Lie group is actually an embedded Lie subgroup. Next we prove a higher-dimensional generalization of the fundamental theorem on flows: if G is a simply connected Lie group, then any Lie algebra homomorphism from its Lie algebra into the set of complete vector fields on a smooth manifold M generates a smooth action of G on M. Using this theorem, we prove that there is a one-to-one correspondence between isomorphism classes of finite-dimensional Lie algebras and isomorphism classes of simply connected Lie groups. At the end of the chapter, we show that connected normal subgroups of a Lie group correspond to ideals in its Lie algebra.