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By Manin Yu.I.

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Example text

Of Chap. I where d/dxt ffilk differentiation. 8tt<'> and 8/> = by s e t t i n g d(8«(»)=8«(M-i>. 8 , s i n c e [8, d] i s t r i v i a l on t h e u\^ . lift I t i s not t h e v a l i d i t y of t h e f o l l o w i n g formal v e r s i o n of Theorem 4 . 1 of Chap. I . 5. LEMMA. P&3B there exists a unique representation of SP in the form For any N-i 2 a ^ S 1 ^ ) ; for t h i s r e p r e s e n t a t i o n Q M + ^> Qlr°=6A . l i-0 ( A l l formulas of S e c . 4 , Chap. 6. w 1 2u £ ' 0) We d e f i n e t h e r e p l a c e d by S3((£"'))-module deriva- Sw^' ) .

X=uN\ -\-0(\ ~ ) 1 Obviously, . *=c+0(\N~r-2) and its uniqueness up toO(r(,+1)) . We X,e33((Z- )), such that Xr = wl+0(\), seek Xr+l in the form X„x = Xr + xr+iZ~lr+1) . Using the distributivity of composition, we have Af-l (X, + x r + 1 r ( r + I >)°" = X°N + 2 K'°x'r+t~(,+1)=Jc;<^-'-" + (remainder). -l We compute the terms on the r i g h t up to N r 3) 0(\ ~ ~ ) . The sum has the form 2 wx ' r+i' w"-l-^N~r~'2 + 0(lN~r~3) . The remainder consists of a sum of products of j

Z£, as in Sec. 2 for dxL= 2 Ci W - ( Lw ) • r ff *=o L<*-o We shall recall that they now do not &\d\, $ = Af,($30)> $o = *l a i%l- A representation analogous to (7) holds as before but only up to a commutator in the term dxUQ, which dropped out for /v-i We S^QN . 1 = 1: 1 AT-1 p-i i J L ' The contribution from this commutator will vanish if we go over to equations for the matrix traces of the Uk in place of the Uk themselves. This becomes even more necessary if we wish to carry over to the matrix case at least a part of the results regarding the Hamiltonian property.

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Algebraic aspects of nonlinear differential equations by Manin Yu.I.

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