Involutive set of functions

Lie Algebras Complete involutive sets of functions. Orbits of general position. Complete involutive sets of functions. Semisimple (singular) orbits. The remaining singular orbits. 

Complete involutive sets of functions. Orbits of general position. 

Ginzburg (1980). Complete involutive set of functions, consists of polynomials. This set is invariant under a coad-joint group representation (but a separate polynomial is, of course, not invariant). 

Fomenko and Trofimov have put the following question when do complete involutive sets of functions (that is, maximal linear commutative algebras of functions) exist on a dual space G and what is the way to find them It turned out that in this problem the leading role is played by FVobenius algebras. But as has been notified by Brailov, the function extension method exploited in this case may be formulated without the assumption that the algebra A possesses FVobenius properties. 

Le Ngok Tyeuen. Complete involutive sets of functions on extensions of Lie algebras connected with Frobenius algebras. In Trudy Seminara po Vect. i Tenz, AnaLf issue 22 (1985), 69-106. Moscow, Moscow Univ. Press. 

The author planned the program of the study of maximal involutive sets of functions on the orbits of Lie algebras. Investigations in this direction were started and first discussed at the above-mentioned seminar in 1980. The most interesting results obtained are also included in the book. 

Integrating a given system in the sense of Liouville means involving the system s Hamiltonian / in the family of functions which are in involution and are such that among them one can choose n independent functions, where n is half the dimension of the enveloping manifold. If such a set of functions is found, then (under the assumptions of Theorem 1.2.5) the trajectories of the system move along tori of half the dimension and set on these tori a conditionally periodic motion in appropriate coc pdinates. 

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. 

Corollary 4.2.1. Let G be a semisimple complex Lie algebra, a an element of general position, V an involutive (commutative) set of functions on the algebra G... 

Theorem 4.4.5 (Brailov). If A is a commutative, associative, Firobenins algebra with unity and /i, ., /m complete involutive set of independent functions on G, then the set of extended functions complete... 

Theorem 5.3.4. If dim Hi (M, Q) > dim Af , then the geodesic Sow on a closed analytic manifold Af does not possess a complete involutive set of analytic functionally independent Srst integrals. 

Richard Dedekind s modularity law holds for closed subsets. The treatment of so-called length functions, which one obtains from generating sets of closed subsets, is another subject which allows one to mimic group theoretic techniques in scheme theory. Finally, the notion of a closed subset leads us to the notion of an involution in scheme theory. 

It is well known that if a smooth function / with nondegenerate critical points, i.e., a Morse function, is given on a smooth manifold Q, then knowing these points and their indices allows us to say much about the topology of the manifold Q. It will be shown in the present chapter that an analogue of this theory exists also in the case where on a symplectic manifold a set of independent functions in involution is given, the number of which is equal to half the dimension of the manifold. 

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... 

Definition 3.4.3 The sympleetic structure (A/,o ) is called meromorphically (respectively, additively) integrable if on Af there exists n independent meromorphic (respectively, additive) functions hi,..., which are in involution with respect to the Poisson bracket and possesses the following property for a certain set of complex numbers (ci,..., c ) G C", the closure of the level set hi = ci,..., hn = Cn does not intersect the divisor of the poles hi,..., hn. 

K 0p V, where K is an arbir trary semisimple Lie algebra (real or complex), p an arbitrary linear representation (reducible or irreducible), dim V being greater than dim K, and in the decomposition of the representation p into irreducible components trivial summands are absent. Brailov, Pevtsova [106], [111]. All the functions of these complete involutive sets are linear. 

Let us now come to the property of integrability in Liouville s sense. To this end we need the following definition a set flq,..., of independent functions is said to be a complete involution system in case it satisfies = 0 for every pair j, k. A classical theorem due to Liouville is the following ... 

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