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Proposition (see [3]). 1). £n — 1, r = 0 , 1 , . . do not contain constants. Then T=0. Thus resolvents are completely determined by their constants in the coef­ ficients TJ0 r,j£n — I. 11 A family of Hamihonian structures related to nonlinear integraUe differential equations Theorem. The set of all resolvents of the operator L is an n-dimensional space over the field C((z -1 )) of series with constant coefficients. The canonical components of the exact resolvents can be taken as a basis. For the proof it suffices to note that for the canonical component TM the constants in T}% r are e,k~m+l when r=n — 1—j and there are no other con­ stants.

Invent Math. SO (3) (1979) 219-248. 6. : Geometry of manifolds. Academic Press, 1964. 7. : A Lie algebra structure in the formal variational calculus, Funkt AnaL 10(1) (1976) 18-25. 8. : The periodic problem for the Korteweg-de Vries equation in the class of finite-gap potentials, Funkt AnaL 9 (3) (1975) 41-52. 25 Reprinted with permission from Sov. Math. Vol. 30, pp. 1975-2036,1984 © 1985 Plenum Publishing Corporation LIE ALGEBRAS AND EQUATIONS OF KORTEWEG-DE VRIES TYPE V. G. Drinfel'd and V. V.

2) contains only the coefficients Xpt, p+q£n — 1, and they must be expanded in series in z - 1 ) . 2). 2) is n-dimensional. 4. We calculate the commutator of the vector fields MX(L) and MY(L) which is denoted as [[MX(L), M,r(L)]]. The commutator is a vector field M(L) such that for any / ( L ) M(L)f(L) = Mx(L)MY(L)f(L)-MY(L)Mx(L)f(L). It the fields depend on L linearly, the rule for calculating the commutator is ttMx(L\Mr(Lm = MY(Mx(L))-Mx(Mr(L)). ) If the fields depend on L polylinearly then one of the fields must be substituted into the other for all the arguments in succession.

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