By Anatoliy K. Prykarpatsky, Ihor V. Mykytiuk (auth.)

In contemporary instances it's been acknowledged that many dynamical structures of classical mathematical physics and mechanics are endowed with symplectic constructions, given within the majority of circumstances by means of Poisson brackets. quite often such Poisson constructions on corresponding manifolds are canonical, which supplies upward push to the potential of generating their hidden team theoretical essence for lots of thoroughly integrable dynamical structures. it's a good understood indisputable fact that nice a part of accomplished integrability theories of nonlinear dynamical structures on manifolds relies on Lie-algebraic principles, by way of which, specifically, the class of such compatibly bi Hamiltonian and isospectrally Lax sort integrable platforms has been performed. Many chapters of this publication are dedicated to their description, yet to our remorse to this point the paintings has now not been accomplished. Hereby our major aim in each one analysed case is composed in setting apart the fundamental algebraic essence answerable for the full integrability, and that's, while, in a few feel common, i. e. , attribute for them all. Integrability research within the framework of a gradient-holonomic set of rules, devised during this booklet, is fulfilled via 3 levels: 1) discovering a symplectic constitution (Poisson bracket) reworking an unique dynamical process right into a Hamiltonian shape; 2) discovering first integrals (action variables or conservation laws); three) defining an extra set of variables and a few practical operator amounts with thoroughly managed evolutions (for example, as Lax kind representation).

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Proof. Fix an element x of the nonempty set R(m) n mo and a Cartan sub algebra I) in 9 containing x and invariant with respect to 7. Let R be the root system of 9 relative to I). Choose eigenvectors Ea E 9 corresponding to the roots a E R such that 7(Ea) = E_ a and ~(Ea, E-a) = -1. Let Ro = {a E R : a(x) = a}. , Ea E e if aERo. 1) in the form (rank 9 - rank eX) + ~(Card R - Card Ro) = dimm. 5) Since 7(X) = x, there exists a basis B for the root system R such that (3(ix) 2': 0 for all {3 E B. Let b the Borel subalgebra defined by the set of positive roots R+ in R relative to B.

X) is polynomial in ).. iVi(x), p i=l +1= degJ. ). iFi(X) , i=o where Fi(x) = grad Vi(X). If G is a connected complex Lie group with Lie algebra g, then lnt (g) = Ad (G) and for f E 1(g) 0= dd t I t=O df(y)[z, y] f(Ad (exptz)y) = * (grad f(y), [z, y]) ([y, grad f(y)], z), where z, y E g. Since f E I(g), the equality [grad f(y), y] = 0 holds for any y E g. 3) [E,FP(x)] = O. , 1979} If x E Fi(x) E gi, i 2: o. ~ then 32 CHAPTER 1 Proof. Let x E b be a regular element of g. Then gX = b and consequently FO(x) E b. *

We show first of all that either ~(x) = ~ or ~I(XI) = ~I. cp(h) , h E ~} (>. ~ 0). 5 u of u = X + V + Xl is (eX n a) ED (efl n ar). 5 U ) = dim(m ED p ED m1), (rank 9 - rank e) + dim(e/eX ) = dim m. 2) and the analogous equality for the spherical pair (9I, er) at Xl we obtain (rank ~(x) + dim~(x)) + (rank ~l(XI) + dim~r(xr)) = 2dim~. 5f+v = ex n a we obtain (rank~tx) +dim~(x)) +rank~l:S dimp. 3) can hold only when ~1(X1) = ~I. Thus either ~l(Xr) = ~1, or ~(x) =~. Let ~(x) ~ ~ and ~1 = gl. 5) to be spherical.