New PDF release: An Introduction to Lie Groups and Symplectic Geometry

By Bryant R.L.

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Extra info for An Introduction to Lie Groups and Symplectic Geometry

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For example, the linear fractional transformations of the last example could just as easily been regarded as a local action of SL(2, R) on R, where the open set U ⊂ SL(2, R)×R is just the set of pairs where cs + d = 0. 8 45 Equations of Lie type. Early in the theory of Lie groups, a special family of ordinary differential equations was singled out for study which generalized the theory of linear equations and the Riccati equation. These have come to be known as equations of Lie type. We are now going to describe this class.

In other words in order to find the fundamental solution of a Lie equation for G when the particular solution with initial condition g(0) = m ∈ M is known, it suffices to solve a Lie equation in Gm ! This observation is known as Lie’s method of reduction. It shows how knowledge of a particular solution to a Lie equation simplifies the search for the general solution. ) Of course, Lie’s method can be generalized. If one knows k particular solutions with initial values m1 , . . ,mk = Gm1 ∩ Gm2 ∩ · · · ∩ Gmk .

21 1 6 m m cij cm xm ⊗ xi ∧ xj ∧ xk . k + cjk ci + cki cj 32 Exercise Set 2: Lie Groups 1. Show that for any real vector space of dimension n, the Lie group GL(V ) is isomorphic to GL(n, R). ) 2. Let G be a Lie group and let H be an abstract subgroup. Show that if there is an open neighborhood U of e in G so that H ∩ U is a smooth embedded submanifold of G, then H is a Lie subgroup of G. 3. Show that SL(n, R) is an embedded Lie subgroup of GL(n, R). ) 4. Show that O(n) is an compact Lie subgroup of GL(n, R).

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