The automorphism groups of foliations with transverse linear connection

2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Nina Zhukova ◽  
Anna Dolgonosova
Keyword(s):  
Automorphism Group ◽  
Lie Group ◽  
Automorphism Groups ◽  
Linear Connection ◽  
The Other ◽  
Riemannian Foliations ◽  
Infinite Dimensional ◽  
Infinite Dimensional Lie Group

AbstractThe category of foliations is considered. In this category morphisms are differentiable maps sending leaves of one foliation into leaves of the other foliation. We prove that the automorphism group of a foliation with transverse linear connection is an infinite-dimensional Lie group modeled on LF-spaces. This result extends the corresponding result of Macias-Virgós and Sanmartín Carbón for Riemannian foliations. In particular, our result is valid for Lorentzian and pseudo-Riemannian foliations.

scholarly journals Finite and infinite dimensional Lie group structures on Riordan groups

2017 ◽  
Vol 319 ◽  
pp. 522-566 ◽  
Author(s):  
Gi-Sang Cheon ◽  
Ana Luzón ◽  
Manuel A. Morón ◽  
L. Felipe Prieto-Martinez ◽  
Minho Song
Keyword(s):  
Lie Group ◽  
Infinite Dimensional ◽  
Infinite Dimensional Lie Group ◽  
Group Structures

scholarly journals A SURVEY ON THE AUTOMORPHISM GROUPS OF THE COMMUTING GRAPHS AND POWER GRAPHS

2019 ◽  
pp. 729
Author(s):  
Mahsa Mirzargar
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
The Other ◽  
Power Graph ◽  
Commuting Graph ◽  
Vertex Set ◽  
Delta G ◽  
Nite Group

Let G be a nite group. The power graph P(G) of a group G is the graphwhose vertex set is the group elements and two elements are adjacent if one is a power of the other. The commuting graph \Delta(G) of a group G, is the graph whose vertices are the group elements, two of them joined if they commute. When the vertex set is G-Z(G), this graph is denoted by \Gamma(G). Since the results based on the automorphism group of these kinds of graphs are so sporadic, in this paper, we give a survey of all results on the automorphism group of power graphs and commuting graphs obtained in the literature.

scholarly journals On the Automorphism Group of a Holomorphic Fiber Bundle over A Complex Space

1970 ◽  
Vol 37 ◽  
pp. 91-106 ◽  
Author(s):  
Hirotaka Fujimoto
Keyword(s):  
Automorphism Group ◽  
Lie Group ◽  
Fiber Bundle ◽  
Complex Space ◽  
Automorphism Groups ◽  
Compact Complex Manifold ◽  
Compact Open Topology ◽  
Complex Spaces ◽  
Holomorphic Fiber Bundle ◽  
Holomorphic Automorphisms

In [8], A. Morimoto proved that the automorphism group of a holomorphic principal fiber bundle over a compact complex manifold has a structure of a complex Lie group with the compact-open topology. The purpose of this paper is to get similar results on the automorphism groups of more general types of locally trivial fiber spaces over complex spaces. We study automorphisms of a holomorphic fiber bundle over a complex space which has a complex space Y as the fiber and a (not necessarily complex Lie) group G of holomorphic automorphisms of Y as the structure group (see Definition 3. l).

scholarly journals A few remarks on invariable generation in infinite groups

2020 ◽  
pp. 1-22
Author(s):  
Gil Goffer ◽  
Gennady A. Noskov
Keyword(s):  
Topological Group ◽  
Automorphism Group ◽  
Lie Group ◽  
Automorphism Groups ◽  
Transcendence Degree ◽  
Linear Groups ◽  
Infinite Groups ◽  
Prime Field ◽  
Regular Tree

A subset [Formula: see text] of a group [Formula: see text] invariably generates [Formula: see text] if [Formula: see text] is generated by [Formula: see text] for any choice of [Formula: see text]. A topological group [Formula: see text] is said to be [Formula: see text] if it is invariably generated by some subset [Formula: see text], and [Formula: see text] if it is topologically invariably generated by some subset [Formula: see text]. In this paper, we study the problem of (topological) invariable generation for linear groups and for automorphism groups of trees. Our main results show that the Lie group [Formula: see text] and the automorphism group of a regular tree are [Formula: see text], and that the groups [Formula: see text] are not [Formula: see text] for countable fields of infinite transcendence degree over a prime field.

Lie groups of 𝐶𝑘-maps on non-compact manifolds and the fundamental theorem for Lie group-valued mappings

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hamza Alzaareer
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Fundamental Theorem ◽  
Smooth Manifold ◽  
Rough Boundary ◽  
Infinite Dimensional ◽  
New Class ◽  
Infinite Dimensional Lie Group ◽  
Group Structures ◽  
Infinite Dimensional Lie Groups

Abstract We study the existence of Lie group structures on groups of the form C k ⁢ ( M , K ) C^{k}(M,K) , where 𝑀 is a non-compact smooth manifold with rough boundary and 𝐾 is a, possibly infinite-dimensional, Lie group. Motivated by introducing this new class of infinite-dimensional Lie groups, we obtain a new version of the fundamental theorem for Lie algebra-valued functions.

scholarly journals FUNCTORIAL ASPECTS OF THE RECONSTRUCTION OF LIE GROUPOIDS FROM THEIR BISECTIONS

2016 ◽  
Vol 101 (2) ◽  
pp. 253-276 ◽  
Author(s):  
ALEXANDER SCHMEDING ◽  
CHRISTOPH WOCKEL
Keyword(s):  
Lie Groups ◽  
Present Article ◽  
Lie Group ◽  
Lie Groupoids ◽  
Infinite Dimensional ◽  
Lie Groupoid ◽  
Equivalence Of Categories ◽  
Infinite Dimensional Lie Group

To a Lie groupoid over a compact base $M$, the associated group of bisection is an (infinite-dimensional) Lie group. Moreover, under certain circumstances one can reconstruct the Lie groupoid from its Lie group of bisections. In the present article we consider functorial aspects of these construction principles. The first observation is that this procedure is functorial (for morphisms fixing $M$). Moreover, it gives rise to an adjunction between the category of Lie groupoids over $M$ and the category of Lie groups acting on $M$. In the last section we then show how to promote this adjunction to almost an equivalence of categories.

STOCHASTIC EQUATIONS AND QUASI-INVARIANCE ON INFINITE PRODUCT GROUPS

1999 ◽  
Vol 02 (02) ◽  
pp. 283-288 ◽  
Author(s):  
SERGIO ALBEVERIO ◽  
ALEXEI DALETSKII
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Lie Group ◽  
Existence And Uniqueness ◽  
Infinite Product ◽  
Stochastic Equations ◽  
Compact Lie Group ◽  
Infinite Dimensional ◽  
Countable Power ◽  
Infinite Dimensional Lie Group

A stochastic differential equation on an infinite-dimensional Lie group G constructed as the countable power of a compact Lie group G is considered. The existence and uniqueness of the solutions and quasi-invariance of their distribution are proved.

Lie Groups of Measurable Mappings

2003 ◽  
Vol 55 (5) ◽  
pp. 969-999 ◽  
Author(s):  
Helge Glöckner
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Measure Space ◽  
Infinite Dimensional ◽  
Direct Sum ◽  
Infinite Dimensional Lie Group ◽  
New Construction ◽  
Locally Convex ◽  
Infinite Dimensional Lie Groups ◽  
Measurable Mappings

AbstractWe describe new construction principles for infinite-dimensional Lie groups. In particular, given any measure space (X; Σ, μ) and (possibly infinite-dimensional) Lie group G, we construct a Lie group L∞(X; G), which is a Fréchet-Lie group if G is so. We also show that the weak direct product of an arbitrary family (Gi)i∈I of Lie groups can be made a Lie group, modelled on the locally convex direct sum .

scholarly journals The Lie group of vertical bisections of a regular Lie groupoid

2020 ◽  
Vol 32 (2) ◽  
pp. 479-489
Author(s):  
Alexander Schmeding
Keyword(s):  
Lie Groups ◽  
Gauge Group ◽  
Lie Group ◽  
Group Structure ◽  
Lie Groupoids ◽  
Gauge Groups ◽  
Infinite Dimensional ◽  
Lie Groupoid ◽  
Regularity Properties ◽  
Infinite Dimensional Lie Group

AbstractIn this note we construct an infinite-dimensional Lie group structure on the group of vertical bisections of a regular Lie groupoid. We then identify the Lie algebra of this group and discuss regularity properties (in the sense of Milnor) for these Lie groups. If the groupoid is locally trivial, i.e., a gauge groupoid, the vertical bisections coincide with the gauge group of the underlying bundle. Hence, the construction recovers the well-known Lie group structure of the gauge groups. To establish the Lie theoretic properties of the vertical bisections of a Lie groupoid over a non-compact base, we need to generalise the Lie theoretic treatment of Lie groups of bisections for Lie groupoids over non-compact bases.

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