Maximal linear commutative subalgebra

Definition 3.1.1 We will say that on a sympletic manifold a maximal linear commutative subalgebra of functions Go is given (in the Lie algebra C (M) with respect to the Poisson bracket) if dimGo = n and if in Go one can choose an additive basis consisting of n functions /i,..., /n functionally independent on (almost everywhere). Such an algebra of functions will be sometimes called a complete involutive (commutative) set of functions. 

Let a sympletic manifold be fixed. It is not clear a priori whether on this manifold there exists at least one maximal linear commutative subalgebra of functions, that is, whether there exists atleast one integrable system on M. We shall specify the formulation. 

If is a smooth manifold, then one can always find at least one maximal linear commutative subalgebra ( o It is constructed in a very simple way. It turns out that on C M) a closed 2n-dimensional ball in which the canonical sympletic coordinates Pi, 9i,.. iPm 9n given, one can always construct a set of n independent smooth functions which are in involution and vanish on... 

It is easy to calculate that the functions measure zero, and sewing the functions constructed above, we obtain just the maximal linear commutative subalgebra on M. 

It is not yet clear whether on any of the symplectic manifolds of the four classes listed above there exists a corresponding maximal linear commutative (MLC) subalgebra of functions. 

Theorem 4.1.7 (Trofimov [130]-[133]). Let G be a simple Lie algebra of one of the following types so(n),su(n),sp(n),G2. Then on each orbit of general position in the real form of the Borel (solvable) subalgebra BG (of the algebra G) there always exists a maximal linear commutative algebra of polynomials. These polynomials are written by explicit formulae. 

In this theorem, we did not imply any concrete Hamiltonian system, but described the properties of the whole class of fields of the form sgrad h generated by the annihilator of the covector of general position. A particular case of Theorem S.1.1 is, of course, the classical Liouville theorem. Indeed, if the maximal linear subalgebra of functions G is commutative then its index r = ind G is equal to its dimension k and, theorefore, the maximality condition becomes A + A = dim Af = 2n, that is A = n. In this case, all the tori 17 from Theorem 3.1.1 are ordinary n-dimensional Liouville tori. 

Lemma 3.1.1. If T is a commutative linear subalgebra (not necessarily maximal) in whose additive basis forms functions /i,..., /n independent on... 

Theorem 4.1.9. Let BG be a Borel subalgebra in a semisimple complex Lie algebra G. Then on BG there always exists a maximal commutative linear algebra of polynomials. 

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