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Towards a Lie theory of locally convex groups. (English) Zbl 1161.22012
The article under review reports on the state of the art in the theory of Lie groups modelled on locally convex spaces. Of particular interest in this theory is the interplay between Lie groups (the global objects of the theory) and their Lie algebras (the infinitesimal objects). In contrast to the finite-dimensional case, considering local Lie groups (the local objects) is also of importance for a general understanding. The thoroughly written article aims at explaining these three concepts, along with its fundamental examples, and at describing the results that allow to translate between them. In addition, a long list of open problems is given, indicating many directions for further research.
In general, the passage from the global to the local level is given by restriction and the passage from the local to the infinitesimal level is given by differentiation at the identity element. This comprises the Lie functor and the core of the theory consists in determining how much this functor forgets and how much can be reconstructed from the infinitesimal and the local level. After a long introduction, referring also to the historical development, the subsequent sections of the article are devoted to different classes of locally convex Lie groups and Lie algebras and describe how much of the theory is known for each of them.
Section 1 digresses from the above described aims into a review of the theory of differential calculus on (not necessarily complete) locally convex spaces and the corresponding concept of locally convex manifolds. All important (and sometimes quite subtle) concepts for a good understanding of the remaining text, such as the Fundamental Theorem of Calculus, weak integrals, vector fields and differential forms, are introduced.
Section 2 deals with the general setup of locally convex Lie groups and their associated Lie algebras. The approach taken here is to define a locally convex Lie group to be a group, endowed with a locally convex manifold structure, such that the group operations are smooth. The corresponding Lie algebra is defined to be the Lie subalgebra of left invariant vector fields on this manifold. In addition, the concept of local Lie group and its Lie algebra are defined. Moreover, the first big class of examples, mapping groups, are introduced.
Section 3 is devoted to the most fundamental concept of interplay between Lie groups and Lie algebras, namely regularity. Regular Lie groups are required to possess solutions to particular initial value problems of ODE’s, determined by Lie algebra-valued curves. These Lie groups share many intuitive phenomena with finite-dimensional Lie groups. For instance, they allow to integrate Lie algebra homomorphisms to Lie group homomorphisms (to the corresponding simply connected cover), they possess exponential functions and allow for a Lie group-valued version of the Fundamental Theorem of Calculus. So far, all known Lie groups, modelled on Mackey-complete spaces, are regular and it is one of the fundamental problems of the theory to determine whether this is a theorem or if there exist counterexamples.
Section 4 deals with locally exponential Lie groups. These are, by definition, Lie groups that have an exponential function, restricting to a diffeomorphism on some zero neighbourhood of the Lie algebra. One big class of examples are Banach-Lie algebras and a typical example of a non-locally exponential but regular Lie group is the diffeomorphism group of a compact manifold. Since the exponential function allows to express the group multiplication in terms of the Lie bracket in the BCH series, there is a close relation to BCH-Lie algebras. As regular ones, locally exponential Lie groups allow a similar integration mechanism of Lie algebra homomorphisms, which is slightly weaker since the target group has to be regular. Moreover, locally exponential Lie groups and BCH-Lie groups allow for more structure on closed subgroups and quotient groups than infinite-dimensional Lie groups do in general, which is exposed in some detail. The remainder of the section concerns integral subgroups, i.e., subgroups of (locally exponential or BCH-) Lie groups induced from inclusions of subalgebras.
Section 5 treats the extensions theory of Lie groups and Lie algebras. After exposing the general ideas and some of the fundamental examples, the cohomology theory of Lie group and Lie algebra extensions are developed. This is done in a rather explicit fashion (largely without using tools from homological algebra), which allows to incorporate smoothness and continuity assumptions on the corresponding cocycles and coboundaries directly. Seemingly, this is done since the corresponding categories, in which the homological algebra would take place, fail to be abelian in this smooth setting (for the same reasons as in the topological setting). One of the main advantages that this direct approach to cohomology allows is to set up a Lie functor between Lie group cohomology and Lie algebra cohomology and one may ask similar questions as for the Lie group-Lie algebra relation now also on the level of cohomology. For instance, it is a reasonable and interesting question to determine which Lie algebra cocycles come from Lie group cocycles (or ”integrate” to Lie group cocycles), and which do not. This can be detected with the aid of vanishing theorems for the so called ”period homomorphisms”, which are associated to Lie algebra cocycles and, moreover, to arbitrary differential forms on Lie groups. Typically, the period homomorphism maps some homotopy group of the Lie group in question to some abelian group and a typical theorem asserts that a Lie algebra cocycle integrates to a Lie group cocycle if the associated period homomorphisms vanish. It thus may be understood as a coupling of algebraic (differential) information, coming from the Lie algebra, to topological (global) information from the Lie group.
Section 6 explains the problem of integrating a given Lie algebra to some Lie group. That this works in finite dimension is ensured by Lie’s Third Theorem, which fails in infinite dimensions. This, probably most fundamental difference to finite-dimensional Lie theory, was first observed by W. T. van Est and T. J. Korthagen in [Nederl. Akad. Wet., Proc., Ser. A 67, 15-31 (1964; Zbl 0121.27503)] and may nicely be shown with techniques from the previous section. In general, a Lie algebra is called integrable if it is (isomorphic to) the Lie algebra of a local Lie group and it is called enlargeable if this local group, in turn, comes from a global Lie group. The phenomena that may occur when integrating Lie algebras are delicate and versatile, for some Lie algebras fail to be integrable (like complexifications of Lie algebras of vector fields, cf. L. Lempert [Contemp. Math. 205, 169–176 (1997; Zbl 0887.22008)]) and others are integrable but not enlargeable (as the above counterexample of van Est and Korthagen). The section explains these phenomena, calculates explicitly some images of period maps and also describes some ad-hoc criteria for (non-) enlargeability and -integrability. Most of the time, the integration problem for locally exponential Lie algebras is considered, which allows for a systematic treatment by the period methods, mentioned above. The last part of the section treats non-locally exponential Lie algebras, for which no such general theory exists.
Section 7 is devoted to direct limits of Lie groups, one of the big classes of examples of infinite-dimensional Lie groups (the other classes, which have already appeared so far, roughly divide into unit groups of Banach algebras, mapping groups and groups of diffeomorphisms). This subject divides itself roughly into direct limits of finite-dimensional groups, for which a good amount of theory has been established by H. Glöckner [Compos. Math. 141, 1551–1577 (2005; Zbl 1082.22012)], and more generally direct limits of infinite-dimensional Lie groups, for which only case-by-case treatments exist.
Section 8 treats (locally exponential Lie subgroups of) unit groups of continuous inverse algebras (CIAs), which are natural generalisations of (subgroups of) unit groups of Banach algebras. These Lie groups are the proper replacement for linear groups in finite dimensions. Some examples and properties are discussed.
Section 9 considers actions of infinite-dimensional Lie groups. Some surprising phenomena occur here, for instance a relatively simple argument implies that there exist no natural Banach-Lie group structures on diffeomorphism groups of compact manifolds. One of the most prominent examples is the action of the diffeomorphism group of some compact manifold \(M\) on the mapping group \(C^\infty(M,G)\) by pre-composition, which is considered in some detail.
Section 10, eventually, is on projective limits of Lie groups. It mostly deals with projective limits of finite-dimensional Lie groups (pro-Lie groups), whose structure theory has recently been developed by K. H. Hofmann and S. A. Morris [The Lie theory of connected pro-Lie groups. EMS Tracts in Mathematics 2. (Zürich): European Mathematical Society (EMS). (2007; Zbl 1153.22006)]. In particular, a version of Lie’s Third Theorem and of the Levi-decomposition for pro-Lie groups are given and local exponentiality and regularity are discussed. A short remark also treats projective limits of infinite-dimensional Lie groups.
The article under review ends in an enormous list of references. It also contains several lists of open problems, so that in conclusion it is the perfect introduction to the research going on in the field.

MSC:
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
22E15 General properties and structure of real Lie groups
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