Groupes différentiels et physique mathématique

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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“…For an approach to Kac-Moody groups in the context of diffeological groups, we refer to [Les03] (cf. [So84]). …”

Section: Theorem Vi27 Let E Be a Locally Convex Space And D ∈ Gl(ementioning

confidence: 99%

Towards a Lie theory of locally convex groups

2006

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“…The notions of diffeology and diffeological spaces are due to Chen [Che] (under the name "differentiable spaces") and to Souriau [Sou1,Sou2]. For details, see [Igl1,Igl2].…”

Section: Diffeological Orbifoldsmentioning

confidence: 99%

Orbifolds as diffeologies

Iglesias

Karshon

Zadka³

2010

Trans. Amer. Math. Soc.

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Abstract. We consider orbifolds as diffeological spaces. This gives rise to a natural notion of differentiable maps between orbifolds, making them into a subcategory of diffeology. We prove that the diffeological approach to orbifolds is equivalent to Satake's notion of a V-manifold and to Haefliger's notion of an orbifold. This follows from a lemma: a diffeomorphism (in the diffeological sense) of finite linear quotients lifts to an equivariant diffeomorphism.

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“…Diffeology as a subject, introduced by Souriau in the 80's [10,11], belongs among various attempts made over the years to extend the usual setting of differential calculus and/or differential geometry. Many of these attempts appeared in the realm of mathematical physics, such as smooth structuresà la Sikorski or a la Frölicher, and were motivated by the fact that many objects that naturally appear in, for example, noncommutative geometry, such as irrational tori, orbifolds, spaces of connections on principal bundles in Yang-Mills theory, and so on, are not smooth manifolds and cannot be easily treated by similar methods.…”

Section: Introductionmentioning

confidence: 99%

Diffeological vector pseudo-bundles

Pervova

2016

Topology and its Applications

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We consider a diffeological counterpart of the notion of a vector bundle (we call this counterpart a pseudo-bundle, although in the other works it is called differently; among the existing terms there are a "regular vector bundle" of Vincent and "diffeological vector space over X" of Christensen-Wu). The main difference of the diffeological version is that (for reasons stemming from the independent appearance of this concept elsewhere), diffeological vector pseudo-bundles may easily not be locally trivial (and we provide various examples of such, including those where the underlying topological bundle is even trivial). Since this precludes using local trivializations to carry out many typical constructions done with vector bundles (but not the existence of constructions themselves), we consider the notion of diffeological gluing of pseudo-bundles, which, albeit with various limitations that we indicate, provides when applicable a substitute for said local trivializations. We quickly discuss the interactions between the operation of gluing and typical operations on vector bundles (direct sum, tensor product, taking duals) and then consider the notion of a pseudo-metric on a diffeological vector pseudo-bundle. MSC (2010): 53C15 (primary), 57R35 (secondary).

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Groupes différentiels et physique mathématique

Cited by 26 publications

References 0 publications

Towards a Lie theory of locally convex groups

Towards a Lie theory of locally convex groups

Orbifolds as diffeologies

Diffeological vector pseudo-bundles

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