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Additional info for An Introduction to Lie Groups and Symplectic Geometry

Example text

22. Show that, for a connected Lie group G, a connected Lie subgroup H is normal if and only if h is an ideal of g. ) 23. For any Lie algebra g, let z(g) ⊂ g denote the kernel of the homomorphism ad: g → gl(g). Use Theorem 2 and Exercise 16 to prove Theorem 4 for any Lie algebra g for which z(g) = 0. ) Show also that if g is the Lie algebra of the connected Lie group G, then the connected Lie subgroup Z(g) ⊂ G which corresponds to z(g) lies in the center of G. ) 24. For any Lie algebra g, there is a canonical bilinear pairing κ: g × g → R, called the Killing form, defined by the rule: κ(x, y) = tr ad(x)ad(y) .

Let M = RP1 , denote the projective line, whose elements are the lines through the origin in R2 . We will use the notation xy to denote the line in R2 spanned by the non-zero vector x y . Let G = SL(2, R) act on RP1 on the left by the formula a b c d · x ax + by = . y cx + dy This action is easily seen to be almost effective, with only ±I2 ∈ SL(2, R) acting trivially. Actually, it is more common to write this action more informally by using the identification RP1 = R∪{∞} which identifies xy when y = 0 with x/y ∈ R and 10 with ∞.

Actually, it is clear that, because of the skew-symmetry of the bracket, only n n2 of these constants are independent. In fact, using the dual basis x1 , . . , xn of g∗ , we can write the expression for the Lie bracket as an element β ∈ g ⊗ Λ2 (g∗ ), in the form β = 12 cijk xi ⊗ xj ∧ xk . 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.

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


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