Cds202-hw6_wi09 - X TG is a left invariant vector eld on G...

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CALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems CDS 202 R. Murray Winter 2009 Problem Set #6 Issued: 12 Feb 09 Due: 19 Feb 09 Reading: Abraham, Marsden, and Ratiu (MTA), Sections 5.1 and 5.2 Problems: 1. MTA 5.1-1: properties of the adjoint map on GL( n ) 2. MTA 5.1-2: tangent group 3. [Warner, page 135, #16; MTA 5.1-5] (a) Let G be a Lie group. Show that the set of right invariant vector fields on G forms a Lie algebra under the Lie bracket operation and that it is naturally isomorphic to T e G . (b) Let φ : G G be the diffeomorphism defined by φ ( g ) = g - 1 . Prove that if
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Unformatted text preview: X TG is a left invariant vector eld on G then * ( X ) is a right invariant vector eld whose value at e is-X ( e ). Further show that X 7 * ( X ) gives a Lie algebra isomorphism of the Lie algebra of left invariant vector elds with the Lie algebra of right invariant vector elds on G . (A Lie algebra isomorphism is a linear mapping A : V V which preserves the Lie bracket: A [ , ] = [ A,A ].) 4. MTA 5.2-1, parts (iii)(v): calculations on SO(3) 5. MTA 5.2-5: the Euclidean group...
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