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By S.P. Novikov, A.T. Fomenko

One carrier arithmetic has rendered the 'Et moi, ..., si j'avait su remark en revenir, je n'y serais element aile.' human race. It has placed logic again Jules Verne the place it belongs, at the topmost shelf subsequent to the dusty canister labelled 'discarded n- sense'. The sequence is divergent; consequently we are able to do whatever with it. Eric T. Bell O. Heaviside Matht"natics is a device for idea. A hugely invaluable instrument in a global the place either suggestions and non linearities abound. equally, all types of components of arithmetic seNe as instruments for different components and for different sciences. making use of an easy rewriting rule to the quote at the correct above one reveals such statements as: 'One carrier topology has rendered mathematical physics .. .'; 'One carrier common sense has rendered com puter technological know-how .. .'; 'One provider classification idea has rendered arithmetic .. .'. All arguably real. And all statements available this manner shape a part of the raison d'etre of this sequence.

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We denote by T(2n, R) the group of phase space translations: T (z0 ) ∈ T(2n, R) is the mapping z −→ z + z0 . Clearly T(2n, R) is isomorphic to Rn ⊕ Rn equipped with addition. Definition 26. The affine (or inhomogeneous) symplectic group is the semi-direct product ASp(2n, R) = Sp(2n, R) T(2n, R). Formally, the group law of the semi-direct product ASp(2n, R) is given by (S, z)(S , z ) = (SS , z + Sz ); this is conveniently written in matrix form as S 01×2n z 1 S 01×2n z 1 = SS 01×2n Sz + z . 7) 22 Chapter 2.

Free Symplectic Matrices Proof. The inverse of SW is the symplectic matrix −1 SW = DT −C T −B T AT which is thus free since det(−B T ) = (−1)n det B. In view of part (i) in Proposition −1 50 the inverse SW is generated by the function W ∗ (x, x ) = − 21 AT (B T )−1 x2 + (B T )−1 x · x − 12 (B T )−1 DT x 2 = − 21 B −1 Ax2 + B −1 x · x − 12 DB −1 x 2 = −W (x , x) (recall that AT (B T )−1 = B −1 A and (B T )−1 D T = DB −1 ). The statement in the following exercise implies that almost every symplectic matrix is free: Exercise 53.

8) of the standard symplectic form can be rewritten in a convenient way using the symplectic standard matrix J= 0 −I I 0 where 0 and I are the n × n zero and identity matrices. In fact σ(z, z ) = Jz · z = (z )T Jz. 2. Symplectic forms 23 Exercise 28. Show that the standard symplectic form is indeed non-degenerate. Let s be a linear mapping Rn ⊕ Rn −→ Rn ⊕ Rn . The condition σ(sz, sz ) = σ(z, z ) is equivalent to S T JS = J where S is the matrix of s in the canonical basis of Rn ⊕ Rn that is, to S ∈ Sp(2n, R).

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