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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. complete-vector field then Exp tX will denote the one-parameter 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
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
1974
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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. complete-vector field then Exp tX will denote the one-parameter 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 .
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Genre | |
Type | |
Language |
eng
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Date Available |
2010-01-29
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Provider |
Vancouver : University of British Columbia Library
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Rights |
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
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DOI |
10.14288/1.0079498
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Campus | |
Scholarly Level |
Graduate
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Aggregated Source Repository |
DSpace
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Item Citations and Data
Rights
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.