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Additional info for Algebra and Trigonometry (International Textbooks in Mathematics)

Example text

2 is finite and hence defines a polynomial map C : g × g −→ g. Indeed, it provides g with a group law: Consider the exponential map exp : g −→ G for the simply connected Lie group with Lie(G) ∼ = g. Then, exp being diffeomorphic near 0 ∈ g, the conditions for a group law are satisfied near the origin and hence everywhere by the identity theorem for polynomial maps (A map V ×V −→ K for a K-vector space V is called polynomial if it is a K-linear combination of products of linear forms in one of both variables.

Then 1 F˙ (t) = f (ad(F (t)))(X + Y ) + [X − Y, F (t)], 2 where f is a convergent power series in t with rational coefficients, indeed f (t) = t t − . −t 1−e 2 37 Together with the initial condition F (0) = 0 ∈ g we obtain then a recursive formula for the homogeneous Lie bracket polynomials Cn (X, Y ). Y ) = [X, Y ] 2 and 1 C3 (X, Y ) = ([[X, Y ], Y ] − [[X, Y ], X]) 12 as well as C4 (X, Y ) = − 8 1 ([Y, [X, [X, Y ]]] + [X, [Y, [X, Y ]]]). 1. A connected (K-Lie) subgroup of a K-Lie group G is a pair (H, ι) with a connected Lie group H together with an injective (K-Lie group) ˜ ˜ι) are equivalent homomorphism ι : H −→ G.

In order to simplify notation we shall from now on identify a connected Lie subgroup (H, ι) with its image ι(H) and its Lie algebra h with the subalgebra ι∗ (h) ⊂ g. 12. e. X ∈ g, Y ∈ h =⇒ [X, Y ] ∈ h. Proof. The subgroup H is normal iff κa (H) = H for all a ∈ G. Obviously that implies Te κa (h) = h for all a ∈ G, in particular for a = exp(X) with X ∈ g. So h Te (κexp(X) )(Y ) = Ad(exp(X))(Y ) = ead(X) Y gives once again, after replacing X with sX, s ∈ K and differentiation with respect to s at s = 0 that d s·ad(X) (e Y )s=0 = ad(X)(Y ) = [X, Y ], h ds the subalgebra h ⊂ g being closed in g.