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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 .

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