An Introduction to Lie Groups and Symplectic Geometry by Bryant R.L.

By Bryant R.L.

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Then any Lie equation for G can be solved by a sequence of quadratures. A Lie algebra with the property described in Proposition 4 is called “solvable”. For example, the subalgebra of upper triangular matrices in gl(n, R) is solvable, as the reader is invited to check. While it may seem that solvability is a lot to ask of a Lie algebra, it turns out that this property is surprisingly common. The reader can also check that, of all of the two and three dimensional Lie algebras found in Lecture 2, only sl(2, R) and so(3) fail to be solvable.

The matrix of 2-by-2 minors. Thus, A−1 = (det(A))−1 tAdj(A). ) It then follows that ( [y2 , y3 ] [y3 , y1 ] [y1 , y2 ] ) = ( y1 y2 y3 ) C , where C = A−1 C Adj(A) = det(A) A−1 C tA−1 . It follows without too much difficulty that, if we write C = S + a ˆ, where S is a symmetric 3-by-3 matrix and   1  a 0 −a3 a2 where a =  a2  , 0 −a1  ˆa =  a3 −a2 a1 0 a3 then C = S + a , where S = det(A) A−1 S tA−1 and a = tAa. Now, I claim that the condition that the Jacobi identity hold for the bracket defined by the matrix C is equivalent to the condition Sa = 0.

Given a homomorphism of Lie groups φ: H → G, define a smooth left action λ: H ×G → G by the rule λ(h, g) = φ(h)g. Then He = ker(φ) and H ·e = φ(H) ⊂ G. In particular, Theorem 1 implies that the kernel of a Lie group homomorphism is a (closed, normal) Lie subgroup of the domain group and the image of a Lie group homomorphism is a Lie subgroup of the range group. Example 3. Any Lie group acts on itself by conjugation: g · g0 = gg0g −1 . This action is neither free nor transitive (unless G = {e}). Note that Ge = G and, in general, Gg is the centralizer of g ∈ G.

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