A topological conjugacy of invariant flows on some class of Lie groups (original) (raw)
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On topological conjugacy of left invariant flows on semisimple and affine Lie groups
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In this paper, we study the flows of nonzero left invariant vector fields on Lie groups with respect to topological conjugacy. Using the fundamental domain method, we are able to show that on a simply connected nilpotent Lie group any such flows are topologically conjugate. Combining this result with the Iwasawa decomposition, we find that on a noncompact semisimple Lie group the flows of two nilpotent or abelian fields are topologically conjugate. Finally, for affine groups G = HV , V ∼ = n , we show that the conjugacy class of a left invariant vector field does not depend on its Euclidean component.
Transitivity of Families of Invariant Vector Fields on The Semidirect Products of Lie Groups
Transactions of the American Mathematical Society, 1982
In this paper we give necessary and sufficient conditions for a family of right (or left) invariant vector fields on a Lie group G to be transitive. The concept of transitivity is essentially that of controllability in the literature on control systems. We consider families of right (resp. left) invariant vector fields on a Lie group G which is a semidirect product of a compact group K and a vector space V on which K acts linearly. If 5F is a family of right-invariant vector fields, then the values of the elements of if at the identity define a subset T of 7.(0) the Lie algebra of G. We say that if is transitive on G if the semigroup generated by U XE¡,{exp(tX): t » 0} is equal to G. Our main result is that if is transitive if and only if Lie(F), the Lie algebra generated by T, is equal to L(G).
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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: isv a. complete-vector: field-then; Exp tX will denote the one-parameter group of X. A local action <£ of G oh 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 : (BI) 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 r (0) of radius r at 0 such that Exp : L-> D(M) is infective on B r (0). Conversely, if L is a suba-lgebra of V(M) (M Hausdorff) satisfying conditions (A) and (B) we show there exists a unique connected Lie transfor-+ mation group with natural action : G x M-> M such that tj) is a Banach Lie algebra isomorphism of L(G) onto L .
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