Some integrable subalgebras of the Lie algebras of infinite-dimensional Lie groups

Abstract

Abstract.This paper gives a proof of Lie's second fundamental theorem in the context of infinite dimensional Lie groups; that is, we define a class of Lie subalgebras of the Lie algebra of a large class of infinite dimensional Lie groups, say G, which can be realized as the Lie algebras of Lie subgroups of G .

Towards a Lie theory of locally convex groups

“…As F is topologically isomorphic to R N , each closed subspace is complemented ([HoMo98, Th. 7.30(iv)]), so that the existence of the corresponding integral subgroup could also be obtained by the methods developed in [Les92,Sect. 4] which require complicated assumptions on groups and Lie algebras.…”

Regular Fréchet-Lie groups of invertible elements in some inverse limits of unital involutive Banach algebras

“…Moreover, even with a stronger hypothesis (by considering only closed Lie subalgebras) and with a weaker result (embedding Lie subgroups instead of immersed Lie subgroups), the proof for an extension of Lie's second theorem meets, for non-Banach Lie groups, with major difficulties or obstructions due to the fact that there is no "nice" theory of differential equations, and a fortiori no "good" Frobenius theorem for the most part of non-Banach Hausdorff locally convex topological vector spaces in spite of some efforts using the bornological machinery (see, for example, [10]). …”

“…Although this assertion could be most likely deduced from some results of J. Leslie in [10], the type of proof given here is interesting in itself: the proofs generally used for this theorem in the infinite-dimensional context require appropriate versions of the implicit functions theorem and of Frobenius's theorem (see, for example, [1] for the Banach case, and [10] for more general cases with the use of a bornological machinery); in contrast, our proof is only based on some properties of analytic foliations and on the use of the Campbell-Hausdorff formula.…”

“…(c) Section 3 is related to some aspects of differentiability and analyticity on Fréchet spaces according to the ideas of J. Leslie, J. Bochniack, J. Siciak, and some others (see, for example, [8], [9], [6], [10], [5]) and to the notion of generalized Campbell-Baker-Hausdorff Lie group (for shortness: CBH-Lie group) initiated by J. Milnor in [11], which are not necessarily known by nonspecialists in infinite-dimensional analysis and which we shall use in the next sections.…”

Lie Groups and Lie Algebras I

“…Let It is also known that the group (Diff M)O acts transitively on the connected components of the manifold of symplectic structures and the manifold of contact structures on a compact manifold M, however, it seems to be unknown if the stabilizers of these actions are Lie subgroups of Diff M. Lie subgroups of Lie-Frechet groups are also discussed in (Leslie 1992).…”