Maximal linear subalgebra

It is clear that if G is an MLC subalgebra in G (Af), then it is an ML subalgebra. Generally, the converse is false, first of all because a maximal linear subalgebra may be nonconnimutative. We will soon give such examples (many of them). 

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. 

Now we shall look at Theorem 3.1.1 from the point of view of the possibility of integrating concrete systems v = sgrad AT. By fixing the Hamiltonian H, we set a certain element of a Lie algebra G°°(Af). Consider in this algebra various maximal linear subalgebras G (if they do exist) and try to find at least one such subalgebra that the Hamiltonian H get into G. 

If the Hamiltonian H admits a correct embedding in a certain maximal linear subalgebra G, then defined is a certain (at least one) common nonsingular level surface C M which corresponds to the covector of general position... 

Theorem 3.1.2 is an obvious corollary to Theorem 3.1.1 We should emphasize an important fact not nearly all functions g from the maximal linear subalgebra G are usual integrals of the field t = sgrad H. Integrals (in the usual sense) are only the functions h from the annihilator Ann c G of the covector If the algebra G is noncommutative then g, H Oior g Ann... 

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. 

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

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. 

Definition 3.1.2 Let G be a finite-dimensional subalgebra in a Lie algebra C [M) (with respect to the Poisson bracket). The subalgebra G will be called a maximal linear (ML) subalgebra on a sympletic manifold if dimG + indG = dimM and if we can choose in G an additive basis consisting of functionally independent (almost everywhere) functions on the manifold M. 

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. 

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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