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Not only does the iPod look amazing, it has performance that blows away the competition It can store between 2000 and 4000 tracks depending on the model bought; many MP3 players have 5 per cent or less of this storage capacity And all this in a player the size of a cigarette case! In addition, the connection to the PC uses a different technology to that of most MP3 players (Firewire instead of USB) This means that you can download an album from your PC in seconds rather than minutes (In case you are wondering, I have not been paid by Apple to spout so enthusiastically about its product, it genuinely is that good) The iPod s success also demonstrates the value of a fantastic product in generating free publicity The design and performance of the product meant that happy users became product fans who told and showed everyone about their new purchase In addition, this buzz was boosted by unanimously positive press coverage that recognized the breakthrough nature of Apple s innovation Despite limited advertising investment, the product received good levels of awareness and interest The incredible success of the iPod makes Apple iPod summary you wonder about where Apple could go next After years of ghting the Wintel (Windows 1 It s about the product, stupid and Intel) army of PCs in a battle that is 2 Hot products create buzz worth its seemingly impossible to win, perhaps it should weight in gold look to brand stretch as a better bet Surely 3 Think about the total design conthe breakthrough design of the iPod could cept, including point of sale and be used to inspire a range of other personal outer packaging electronic products, such as portable phones and personal organizers We will now look in more detail at how product, packaging and distribution can help you deliver against your extension promises (If the iPod story has given you enough ideas on this already, skip to the next chapter).

This is the simplest case Given a semisimple Lie group G, its real Lie algebra g always possesses a Cartan involution, that is, an automorphism : g g, with square equal to the identity, [X , Y ] = [ (X ), (Y )] , X,Y g, 2 = I d , (B16)

and such that the symmetric bilinear form B (X , Y ) = B(X , Y ) is positive-definite, where B is the Cartan Killing form Then the Cartan involution yields an eigenspace decomposition

(B17)

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of g into +1 and 1 eigenspaces It follows that [k, k] k , [k, p] p , [p, p] k (B18)

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Assume for simplicity that the center of G is nite, and let K denote the analytical subgroup of G with Lie algebra k Then: K is closed and maximal-compact, there exists a Lie group automorphism of G, with differential , such that 2 = I d and the subgroup xed by is K, the mapping K ~ p G given by (k, X ) k exp X is a diffeomorphism One may also write the diffeomorphism as (X , k) exp X k = p k

Any connected semisimple Lie group G has an Iwasawa decomposition into three closed subgroups, namely, G = K AN , where K is a maximal compact subgroup, A is Abelian, and N nilpotent, and the last two are simply connected This means that every element g G admits a unique factorization g = kan, k K , a A, n N , and the multiplication map K ~ A ~ N G given by (k, a, n) kan is a diffeomorphism Assume that G has a nite center Let M be the centralizer of A in K, that is, M = {k K : ka = ak, a A} (if the center of G is not nite, the de nition of M is slightly more involved) Then P = MAN is a closed subgroup of G, called the minimal parabolic subgroup The interest in this subgroup is that the quotient manifold X = G/P ~ K /M carries the unitary irreducible representations of the principal series of G (which are induced representations), in the sense that these representations are realized in the Hilbert space L2 (X , ), with the natural G-invariant measure To give a concrete example, take G = SO o (3, 1), the Lorentz group Then the Iwasawa decomposition reads SO o (3, 1) = SO(3) A N , (B19)

where A ~ SO o (1, 1) ~ R is the subgroup of Lorentz boosts in the z-direction and N ~ C is two-dimensional and Abelian Then M = SO(2), the subgroup of rotations around the z-axis, so that X = G/P ~ SO(3)/SO(2) ~ S 2 , the 2-sphere A closely related decomposition is the so-called K AK decomposition Let G again be a semisimple Lie group with a nite center, K a maximal compact subgroup, and G = K AN the corresponding Iwasawa decomposition Then every element in G has a decomposition as k 1 ak 2 with k 1 , k 2 K and a A This decomposition is in general not unique, but a is unique up to conjugation A familiar example of a K AK decomposition is the expression of a general rotation SO(3) as the product of three rotations, parameterized by the Euler angles: = m( ) u( ) m( ) , (B20)

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