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Gij / D 0 with FW Rn ! R a polynomial or a rational function of the entries, hence continuous on some open set, their solutions cut out subgroups that are topologically closed. The closed subgroups are very special. 42))). Let G be a Lie group and let H be a closed subgroup of G. Then H has a unique smooth (in fact analytic) structure that makes it a Lie subgroup of G. 43)). Let G and H be two Lie groups with Lie algebra g and h, respectively, and let FW G ! H be a Lie group homomorphism. g/ 2 h.

G/. 3 Representation Theory Let H1 and H2 be two Hilbert spaces (the corresponding norms and scalar product are simply denoted by k k and h ; i). Suppose that AW H1 ! H1 ; H2 /. Recall that A is an isometry if kAuk D kuk for every u 2 H1 . Since kAuk2 D hAu; Aui D hA Au; ui and kuk2 D hu; ui, the polarization identity implies that A is an isometry if and only if A A D idH1 . Hence, isometries are injective, but they are not necessarily surjective. A bijective isometry is called a unitary map. If A is unitary, such is also A 1 and in this case AA D idH2 .

G/Ái D 0 for every g 2 G, contrary to assumption. Hence M D H and is irreducible. 48. 23) of the full affine group is. The calculations that follow are very basic and important. Rd / by Z fO . / D F f . a /; p F . a; b/f /. a /; F . a; b/f /. / D a 2 R ; b 2 R: We start with and show that it is not irreducible. R/. a /Og. a /. b/ˇ2 db da ; D a G jhF . a; b/f /; F gij2 34 F. De Mari and E. a . a /Og. /. a . /j2 d G Z ÂZ O R C1 0 da ; a Ã 2 da O jOg. R/ W fO . R/ W fO . 26), a < 0 for every a > 0.