A Locus with 25920 Linear Self-Transformations by H. F. Baker

By H. F. Baker

Initially released in 1946 as quantity thirty-nine within the Cambridge Tracts in arithmetic and Mathematical Physics sequence, this ebook offers a concise account relating to linear teams. Appendices also are integrated. This booklet might be of price to someone with an curiosity in linear teams and the historical past of arithmetic.

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Recall t h a t a semigroup S with an identity element 1 is called a monoid. 14. Let S be a closed submonoid of the Lie group G. Then the following assertions hold: (i) L(S) is a Lie wedge. (ii) L (H(S)) = g ( L ( S ) ) . Proof. (i) It is dear that IR+ L(S) = L(S). The closedness of L(S) follows from 1 L ( S ) = N t- e x p - ' ( S ) t>O and the continuity of exp. Let X, Y C L(S). To see that L(S) is a wedge we have to show that X + Y E L(S). This follows from exp (t(X + Y)) = limo~ (exp( t-X)exp(t-Y))" ~ S n n Vt E 1R+.

Let G be a connected Lie group with compact Lie algebra 9 = L ( G ) and W C g a Lie wedge. Suppose that H ( W ) is global in G. Then W is global in G iff L(K) fq W C_ H ( W ) for every compact subgroup K C_ G. Proof. 39. To prove sufficiency we notice that the Lie algebra 9 is reductive and therefore 9 = Z(9) • [9,9] where 9' = [9, 9] is a compact semisimple Lie algebra. 180] we find an n E IN and a maximal compact subgroup K C_ G such that G is diffeomorphic to K x JR". Consequently K is connected.

Then W / L ( H ) is global in G / H if and only if W is global in G. Proof. 41. Let p: G --* G / H be the canonical projection and T := p(S). The closedness of T follows from S H = S and the closedness of S C_ G. Let W = L(S) be global in G. Then we find that L(T) = {X e L ( G / H ) : exp(~t+X) C_ T} = { X E L ( G / H ) : p - i (exp(lR+X)) C S} = dp(1){Y E L ( G ) : exp(lR+Y) C_ S} = dp(1) L(S) = W~ L(H). Therefore W~ L(H) is global in G / H . 42 we immediately derive that if G is a connected Lie group, and W C L(G) an invariant wedge such that H ( W ) is global in G, we have that W is global in G if and only if the pointed invariant Lie wedge V := dp(1)W is global in G / H , where p: G ---* G/SH(W) denotes the canonical projection.

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