 Library Home /
 Search Collections /
 Open Collections /
 Browse Collections /
 UBC Theses and Dissertations /
 Lie transformation groups
Open Collections
UBC Theses and Dissertations
UBC Theses and Dissertations
Lie transformation groups Thomson, James P.
Abstract
Suppose G is a Lie group and M is a manifold (G and M are not necessarily finite dimensional). Let D(M) denote the group of diffeomorphisms on M and V(M) denote the Lie algebra of vector fields on M . If X is a. completevector field then Exp tX will denote the oneparameter group of X . A local action ⌽ of G on M gives rise to a Lie algebra homomorphism ⌽⁺ from L(G) into V(M) . In particular if G is a subgroup of D(M) and ⌽ : G x M →M is the natural global action (g,p) → g(p) then G is called a Lie transformation group of M. If M is a Hausdorff manifold and G is a Lie transformation group of M we show that ⌽⁺ is an isomorphism of L(G) onto ⌽⁺ (L(G)) and L = ⌽⁺(L(G)) satisfies the following conditions : (A) L consists of complete vector fields. (B) L has a Banach Lie algebra structure satisfying the following two conditions: (B1) the evaluation map ev : (X,p) → X(p) is a vector bundle morphism from the trivial bundle L x M into T(M), (B2) there exists an open ball B[sub r](0) of radius r at 0 such that Exp : L → D(M) is infective on B[sub r](0). Conversely, if L is a subalgebra of V(M) (M Hausdorff) satisfying conditions (A) and (B) we show there exists a unique connected Lie transformation group with natural action ⌽ : G x M → M such that ⌽⁺ is a Banach Lie algebra isomorphism of L(G) onto L .
Item Metadata
Title 
Lie transformation groups

Creator  
Publisher 
University of British Columbia

Date Issued 
1974

Description 
Suppose G is a Lie group and M is a manifold (G and M are not necessarily finite dimensional). Let D(M) denote the group of diffeomorphisms on M and V(M) denote the Lie algebra of vector fields on M . If X is a. completevector field then Exp tX will denote the oneparameter group of X . A local action ⌽ of G on M gives rise to a Lie algebra homomorphism ⌽⁺ from L(G) into V(M) . In particular if G is a subgroup of D(M) and ⌽ : G x M →M is the natural global action (g,p) → g(p) then G is called a Lie transformation group of M. If M is a Hausdorff manifold and G is a Lie transformation group of M we show that ⌽⁺ is an isomorphism of L(G) onto ⌽⁺ (L(G)) and L = ⌽⁺(L(G)) satisfies the following conditions :
(A) L consists of complete vector fields.
(B) L has a Banach Lie algebra structure satisfying the following two conditions:
(B1) the evaluation map ev : (X,p) → X(p) is a vector
bundle morphism from the trivial bundle L x M into T(M),
(B2) there exists an open ball B[sub r](0) of radius r at 0 such that Exp : L → D(M) is infective on B[sub r](0).
Conversely, if L is a subalgebra of V(M) (M Hausdorff) satisfying
conditions (A) and (B) we show there exists a unique connected Lie transformation group with natural action ⌽ : G x M → M such that ⌽⁺ is a Banach Lie algebra isomorphism of L(G) onto L .

Genre  
Type  
Language 
eng

Date Available 
20100129

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0079498

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.