Introduction Calculus of smooth mappings Calculus of holomorphic and real analytic mappings Partitions of unity Smoothly realcompact spaces Extensions and liftings of mappings Infinite dimensional manifolds Calculus on infinite dimensional manifolds Infinite dimensional differential geometry Manifolds of mappings Further applications References Index.

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The Convenient Setting of Global Analysis

Groupes et algébres de lie

In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. We further describe how regularity or local exponentiality of a Lie group can be used to obtain quite satisfactory answers to some of the fundamental problems. These results are illustrated by specialization to some specific classes of Lie groups, such as direct limit groups, linear Lie groups, groups of smooth maps and groups of diffeomorphisms.

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Towards a Lie theory of locally convex groups

Differential Calculus over General Base Fields and Rings

For general degenerate Poisson brackets, analogues are constructed of invariant vector fields, invariant forms, Haar measure and adjoint representation A pseudogroup operation is defined that corresponds to nonlinear Poisson brackets, and analogues are obtained for the three classical theorems of Lie The problem of constructing global pseudogroups is examined Bibliography: 49 titles

http://iopscience.iop.org/article/10.1070/IM1987v028n03ABEH000895/pdf

Analogues of the objects of lie group theory for nonlinear poisson brackets

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On Regular Fréchet-Lie Groups V; Several Basic Properties

The purpose of this paper is to survey the theory of regular Frechet-Lie groups developed in [1–10]. Such groups appear and are useful in symplectic geometry and the theory of primitive infinite groups of Lie and Cartan [11]. From the group theoretical standpoint, general relativistic mechanics is a more closed system than Newtonian mechanics. Quantized objects of these classical groups are closely related to the group of Fourier integral operators [12]. These can also be managed as regular Frechet-Lie groups. However, there are many Frechet-Lie algebras which are not the Lie algebras of regular Frechet-Lie groups [13]. Thus, the enlargeability of the Poisson algebra is discussed in detail in this paper. Enlargeability is relevant to the global hypoellipticity [14, 15] of second-order differential operators.

The theory of infinite-dimensional lie groups and its applications

where $\varphi=(\varphi_{1};\varphi_{2})$ is a symplectic transformation of order 1 on $T^{*}N-\{0\}$ . Although our operators such as (1) form much narrower class than what was defined by H\"ormander [3] or Guillemin -Sternberg [2], our expression contains less ambiguities, and hence one can give a sort of coordinate system on a “vicinity” of the identity operator of the Fourier-integral operators of order $0$ (cf. Theorem 5.8 [8]). Moreover, the above expression seems to be convenient for concrete computation of the fundamental solution of the equation

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On Regular Fréchet-Lie Groups II; Composition Rules of Fourier-Integral Operators on a Riemannian Manifold

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On regular Fréchet-Lie groups, I some differential geometrical expressions of fourier-integral operators on a Riemannian manifold

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Akira Yoshioka

Papers

On Regular Fréchet-Lie Groups V; Several Basic Properties

The theory of infinite-dimensional lie groups and its applications

On Regular Fréchet-Lie Groups II; Composition Rules of Fourier-Integral Operators on a Riemannian Manifold

On regular Fréchet-Lie groups, I some differential geometrical expressions of fourier-integral operators on a Riemannian manifold

Deformation quantization of Poisson algebras