methods of constructing functions in involution (4s)

THEOREM 3.2. i. If x = x(t), y = y(t) is a solution of the equations ~___a_~,OF ~___~_~OF for

IYl = i, F = 0, then the line ~(x(t), y(t)) is tangent to the surface f(x) = 0, and the point of tangency $(t) of the line ~(x(t), y(t)) with the surface {f = 0} moves along a geodesic of this surface.

4. Methods of Constructing Functions in Involution

4.1. The Method of Shift of Argument. Let f(x) be a function defined on a linear space V, and let aCV be a fixed vector. On the space V we construct a family of functions f~(x)= f(x~-%a), where % is an arbitrary number ( %~R if V is a linear space over R). We say that the functions f%(x) are obtained from f(x) by the operation of shift of argument. If f% can

co

' E n be expanded in % in a series [for example f(x) a polynomial] then f~(x)~- % f~,n(X) and the n=0

shift operation generates an entire family of functions {fz,~(x)} from the function f(x).

In the theory of Hamiltonian systems shift of argument is used because of the following theorem of Mishchenko and Fomenko [54]. This idea for the case of the Lie algebra so(n) ap- peared in the work of Manakov [53].

THEOREM 4.1.1 (see [65]). Let F, H be two functions on the space G* dual to the Lie algebra G which are constant on orbits of the coadjoint representation (are invariants of the coadjoint representation of the Lie group ~ corresponding to the Lie algebra G), let a~G* be a fixed covector, and let ~, ~ER be arbitrary fixed numbers. Then F~(t)=F(t-~-~a) and H~(t)= H(L-~a ) are in involution on all orbits relative to the standard symplectic Kirillov struc- :ture.

This theorem provides a complete involutive collection of functions for a rather broad class of Lie algebras including semisimple Lie algebras (see [63, 65]).

A general construction of shift of basis functions in finite-dimensional representations in the space of smooth functions on G* was proposed in the work of Trofimov [90]; see also the survey [103]. We obtain Theorem 4.1.1 and also the construction of shifts of semiinvariants of the coadjoint representation of the work [4] as a special case of this general construction.

In the work [87] the operation of shift of invariants is generalized to the case of arbitrary finite-dimensional algebras (not necessarily commutative or skew-commutative). A construction of equations of hydrodynamic type for which these shifts are first integrals is also given there.

In the construction of complete involutive families of functions it is also necessary to use nonlinear shifts of argument. We present here one useful assertion of this sort con- cerning nonlinear shifts.

THEOREM 4.1..2 (see [82, 163]). Let G be a Lie algebra equipped with a nondegenerate, G-invariant bilinear form ($, N). Suppose functions f and h satisfy the condition [gradf(~), ~]--0, [grad h(~), ~]=0 for all ~G. We introduce the notation f~(~,~) =~(~-~-~2e), h~(~,B) = h(~-~N~-~2e), where ~ is a fixed element of the Lie algebra G, and % and ~ are arbitrary para- meters. Then the functions [~(~,~) and h~(~,N) are in involution for any % and ~ relative to the canonical Poisson bracket on orbits of the coadjoint representation of the Lie algebra (O) =C| (~[x]/(x~i).

4.2. The Method of Chains of Subalgebras. Suppose in the Lie algebra G there is a filtration of subalgebras ~G~Q~...~Oq. There then arises a chain of mappings O*-+O~-+ O~-+...-+Qq. Each of the mappings Oi-+O~+~ is the restriction to the subalgebra O~+~ of linear functions defined on G i. The following assertion holds.

THEOREM 4.2.1 (see [88, 91]). Suppose there is a chain of subalgebras V m S. If functions f, g on S* are in involution on all orbits of the representation Ad~, where ~ is the Lie group corresponding to the Lie algebra S, then f and g are in involution on all orbits

of the representation Ad~5, where f15 is the Lie group corresponding to the Lie algebra V and

f=foz~, -g-----goz~, ~'V*-+S* is the restriction mapping.

One can become acquainted in more detail with various versions of the method of chains of subalgebras, for example, in the survey [103].

We note that the result of Arkhangel'skii [4] can be strengthened by using Theorem 4.2.1.

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THEOREM 4.2.2. Let F, H be two semiinvariants (F (Adg*x )=x(g )F(x ) , x6G*, g6~J, where ~ is a character of the Lie group ~) of the coadjoint representation of the Lie group ~, corresponding to the Lie algebra G, let =E~* be a fixed covector, and let ~, ~R. Then {f(tq-%a), H(tq-~a)}=O on all orbits of the coadjoint representation relative to the standard symplectic structure.

The assertion of Theorem 4.2.2 follows from Theorems 4.1.1 and 4.2.1, since any semi- invariant is the continuation of an invariant of some smaller subalgebra.~

4.3. The Method of Tensor Extensions of Lie Algebras. This method was first proposed by Trofimov [92, 93] and was then developed by Brailov [i0] and Le Ngok T'euen.

Let G be a Lie algebra, and let K be a commutative ring. Then O | K is a Lie algebra.

In the case where K-----R[xI,.. xn]/(x~ '+I ".,Xmn+1) . . . . n , we denote the Lie algebra G| by f~m ...... m n (G). Let E i denote the image of x i in the ring K.

= ' ~ej, O~<j<r, Let el,...,e r be a basis of the Lie algebra G. Then the vectors el ... 0,.<at~<~! , l<i~<n, form a basis of the Lie algebra ~m ...... mn(U). We denote the dual basis in

G* by el; it is defined from the relation < e l , e] > =6~, where <f, x> is the value of fCO*on

the vector xEU. We denote the coordinates in the spacelQm~,i. ,~n(G)* in the basis dual to the

basis ~'. . .e nane i, l-.<j-~<r, O~at..<r~i l..<i..<Iz, by x(~1 .... . o~n) i, while the coordinates in G* in the

basis dual to el,-l~{-<.-Gwe denote by xi (IKi-.<r). In the space f~,n ....... n(O)* we introduce the new variables

. . . . . (1) o~j+~j=mj . l < j < t z

Description of algorithm (~) of V. V. Trofimov. Let F(xl, .... Xr) be an analytic function on the space G*; we consider the function F(zl, .... z r) on the space f~m ...... mn(G)* with values

in the ring K. By expanding F(x I ..... x r) in a Taylor series and substituting z i from (i), we obtain a finite sum, since sufficiently high powers of Ei are equal to z6ro. Thus, F (z I .... ,z r) is a well defined function on the space ~m,,..~,mn(O)* with values in the ring K; it can be represented in the form

F ( Z l , . . . . Zr)~ ~ 81~I... 8;nF a . . . . ~n (X ([~1 . . . . . ~,~t)t, O~ctj<m]

s i n c e a~l . . . an % i s a b a s i s o f t h e a l g e b r a K~-I~[xl . . . . . Xnl/(x[ ~'+' . . . . . xmn+r), F~ ...... %(X(~31 . . . . . ~n)i)ER. The function F~,...%:Q, n .... mn(G)*.-+kwe call the homogeneous components of the function

F(zl, .... Zr). Algorithm (9~), by definition, takes a function F on the space G* into the collection of homogeneous components ~(F)-----{f=,..=n} of the function F.

THEOREM 4.3.1 (see [93, 96, 97]). Suppose the functions F~(x) ..... FN(X) defined on G* are in involution relative to the Kirillov form on all orbits of the coadjoint representation of the Lie group ~ associated with the Lie algebra G. Then all functions 91(F~)U... U~I(FN) coming out of algorithm (~D are in involution relative' to the Kirillov form on all orbits of the coadjoint representation of the Lie group ~2rnt,..mn(~), associated with the Lie algebra f~m .... mn(O). Moreover, if F~, .... F N are functionally independent on G*, then all functions of

the family ~(FOU... U~(F~)are functionally independent on the space f~m .... an(O)*.

Remark i. If F is an analytic function on an open subset U c G*, then the homogeneous components F~...~n of the function F are also defined on some open subset Lf~f~m .... mn(O)*. The assertion of Theoem 4.3.1 hereby remains valid. This situation arises, for example, in considering rational invariants of the coadjoint representation.

Remark 2. Regarding the connection of the method of tensor extensions with the generalized shift method of the work [90], see [98].

A partial generalization of algorithm (~) to the case of algebras with Poincar4 duality was obtained in the work [i0]. Let A be a graded commutative aigebraA=Ao@...@An, AiA i A~+ i, i + j ~ ~I~.

Definition 4.3.1. Let dimA n = i, and let ~ be a linear . . . . . . . . function on A identically equal to zero on A i for i-.</z--I and not equal to zero on A n . Let ~(g, b)=o(ab) be a syn~metric bilinear form on A. If S is a nondegenerate form, then the algebra A is called an algebra with

Poincar6 duality.

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It is easy to see that for such algebras A 0 = k. Let sl = i. We chose an arbitrary basis ie2 ..... e7, in A I and then choose an arbitrary basis e7,+i ..... 87, in A 2 and so forth to

gradation n/2. In An/2 we choose a basis in which the matrix of the form $ is diagonal. In the spaces Ai, i > n/2 we chose bases dual relative to ~ to those already chosen in An/2_ i . As a result, we obtain Sl, .... SN - a homogeneous basis of the algebra A which is self-adjoint relative to 6: A, ~(e,i, ej)----6~j, where * is permutation.

Let el,...,em be a basis in G*, and let xl,...,x m be the corresponding coordinates. The linear functions xl, .... x m on G* are naturally considered elements of the Lie algebra G. Let (TA-~-O| Then the elements xi| of the Lie algebra G A can be considered linear coordi- nate functions on GI~ Let x{-~x~| be coordinates on G~. For a polynomial function P(x) on G* given by

oo

k=0 1 , . . . , , l k

and I~]~<N we define a polynomial function P(J)(x) on G~:

N

�9 �9 , . 8 . l X? j' "

i , . . . . , t k

J' . . . . . J k

here (ej, sj,.....e&) is the projection of the element ej~....'ei~ onto the basis vector ~j. Let

, and let Ann(x)={gCOIad~x-~-O } be the annihilator of the element x. The dimension of

the annihilator of an element xGfJ* of general position is called the index of the Lie algebra G. Let ~ be a collection of polynomial functions on G*. The collection of polynomial functions on G~ of the form p(i), PC~, l~<j~N, we denote by ~A.

THEOREM 4.3.2 (see [i0]). If ~ is an involutive collection of functions on G*, then ~ A is an involutive collection of functions on G~. If ~ is a complete invoiutive collection and the number of independent polynomial invariants of the Lie algebra G is equal to the index, then ~A is a complete involutive collection on G~. If the collection ~ contains a nondegenerate quadratic form, then the collection,~ ~ also contains a nondegenerate quadratic form.

Le Ngok T'euen proved that the assertion of Theorem 4.3.2 holds only for algebras with " Poincarg duality. More precisely, let G be a Lie algebra, let A be a commutative algebra, let (J| be the tensor pr0duct of G and A, let ~i be a basis of the algebra A, and let be a linear operator, ~:ei--+eo:t, such that ~2 = id. For f~O we set f '~=f|174 ~. If f

and g are polynomials on (O| then we define the function (fg)a by the formula (fg)a ~_~

THEOREM 4.3.3. The equality {f~, gb}={f, g}ab holds if and only if the algebra k is an algebra with Poincar~ duality.

4.4. The Method of Contraction of Lie Algebras. Let G be a Lie algbera. We suppose

that G = H + V whereby I/f, I-/]cH, [/-/,V]cV, IV, V]cff. Then G is called a Zg-graded algebra. Let H*(V*) be the subspace in G* of all covectors annihilating V (respectively, H). Then G*-----H*| *. For any xEO* and gCO everywhere below x H and x V denote the H*- and V*-components of the element x, while gH, gv are the H- and V-components of g. On G we define a new commutator Ix, b']~. by the formula [g, gq~.=[g~, g,v]--]-[gv, gv]-]-[g'v, gv]-[--~,[gv, gv]. The commutator [x, Y]X satisfies the Jacobi identity. The Lie algebra obtained we denote by G k.

Definition 4.4.1. Let G be a Z2-graded Lie algebra, and let G% be the series of Lie algebras constructed above corresponding to the commutators [x, Y]X" The Lie algebra G O is called a contraction of the Z2-graded Lie algebra G.

Let G be a Z2-graded Lie algebra. We denote the Poisson bracket for the contraction G o by {f, g}0- In a manner similar to the way we define the contraction of the commutator [x, y] above, for certain functions F on G* it is also possible to define something like a contraction.

Definition 4.4.2. We suppose that the expansion of F in powers of V breaks off at some

term Fo(xF.)-JrF 1(x~, Xv)~:-... 2rF~ (x~, xK), where F ~(xH, xv) for fixed x H is a k-form in x V. In

this case we set F~(x)=kTF x~, k 2Xv �9 We remark that F~(x) is also defined for X = 0,

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F0(x) = Fn(XH, xv). As A. V. Brailov noted, the operation of contraction of Lie algebras and functions has the following properties.

THEOREM 4.4.1. Let G=H~V be a Z2-graded Lie algebra, and let F, F' be functions on G* such that a) {F, F'} = 0, b) F% and Fi are defined (for example, F and F' are polynomials). It is then asserted that {F~, F~} = 0, and, in particular, {F n, Fim}0 = 0, where F n, F 'm is the maximal component in the expansion in homogeneous components in V.

4.5. The Method of Similar Functions. Let H be a semisimple Lie subaigebra of G. We consider the adjoint action of the Lie algebra H in G. The Lie algebra G becomes an H-module. Let V be an H-module in G not containing H. If in V we fix a basis Xl, .... x n, then on H there is defined an n • n matrix of I-forms. The value of this matrix on a vector h~H we denote by W h. Thus, W h is the matrix of the transformation adh:V ~ V in the basis xl,...,x n. The matrix elements mij of the matrix W are I-forms on H. Since the algebra H is semisimple, mij may be considered to belong to H.

Let B be an invariant r-form relative to the action of the algebra H on rows (XI,. .,Xn) , Xz~V. Such forms exist, since H is semisimple. Then B(XW I~, .... XW It) is an H-invariant, i.e., [h,B(XWi~I ...i )#Wtr)]=0, if we consider B(X~v 6, .... xWti) as an element of the tensor algebra S(G) over G. If we consider vectors of G as linear forms on G*, then this can be reformulated in the form of the following assertion.

Proposition 4.5.1. Functions on G* of the form E(XW 4, .... XW It) are in involution with all the linear functions ~ii (I~<i, j~<n).

It may turn out that the H-module G has several submodules VI,...,Vm isomorphic to V.

In this case it makes sense to speak of functions of the form B(XW4, X~WI',...~ X(m)Wt O. As before, functions of this form are in involution with all the linear functions ~ii, I~i, j~<n.

Definition 4.5.1. Functions of the form B(XW 4, .... x(m)w tr) we call canonical H-invariants.

n--1

According to the Caley-Hamilton theorem Wn=~ ai(W)W i.

Definition 4.5.2. Powers of the matrix W are explicitly present in the notation for a canonical H-invariant. We call two canonical H-invariants similar if one can be obtained from the other by replacement of the matrix W n by W i whereby the power i is such that the function ai(W) is not identically zero.

Definition 4.5.3. Let G be a Lie algebra, and let H be a subalgebra of it. We say that the pair (G, H) satisfy the M-condition if the following conditions are satisfied. We suppose that G=H~VI@ .. ~Vh is the decomposition of the H-module G into irreducible submodules. It is required that a) H be a simple Lie subalgebra in G of classical type A n , Bn, C n, Dn; b) either all V i are trivial H-modules or the representation of the Lie a~%ebra corresponding to the submodule V i (i = l,...,k) is equivalent to the minimal representat~an.

There is the following basic theorem due to A. V. Belyaev.

THEOREM 4.5.1. Let G be a Lie algebra, and let H be a subalgebra Of it. We suppose that the pair (G, H) satisfies the M-condition and F i = x'WiBX t is a canonical H-invariant. Then the Poisson bracket {Fi, Fn} of similar functions Fi, F n (X may coincide with X') is equal to zero.

4.6. The Restriction Theorem. Let (M, m) be a symplectic manifold, let h be a smooth function on M, and let sgradh be the vector field dual relative to the form m to the differ- ential dh. Let E be a group acting by symplectic diffeomorphisms on M, and suppose the function h is constant on orbits of the action of the group E. We then say that E is a group of symmetries of the Hamiltonian system (M, m, h).

There is the following assertion due to A. V. Brailov which is useful in the construction of functions in involution.

THEOREM 4.6.1. Suppose Z is a compact group of symmetries of the Hamiltonian system (M, m, h); ~ is a E-invariant algebra of integrals, ~z is a E-fixed subalgebra, N is the mani-

fold of fixed points, and ~ is the set of restrictions of integrals in ~ to N. It is then asserted that a) (N, ~) is a symplectic manifold where ~ = m/N; b) the set ~ is closed

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relative to the Poisson bracket {f, g}N, and the restriction mapping is an epimorphism of Lie

algebras (~z,{f,g}N)-+~,{f,g}~); c) ~ is an algebra of integrals of the Hamiltonian system (N, ~, h), where h is the restriction of the function h to N.

Symplectic actions of groups arise, for example, in the following situation (A. V. Brai- lov).

Proposition 4.6.1. Let Z be a compact group acting by automorphisms on the Lie algebra G, and let G n be the fixed subalgebra; let xEGn*cG*, let Oa(x) be the orbit of the representation Ad* of the Lie group corresponding to G passing through x, and let Gan(x ) be the orbit

for G n, Then a) Z acts by symplectic diffeomorphisms on Go(x); b) Gon(x ) is open in the manifold of E-fixed points of the orbit GQ(X); c) if ~, mn are the Kirillov forms of the Lie algebras G and Gn, then ~n=~iOn ~.

CHAPTER 3

COMPLETE LIOUVILLE INTEGRABILITY OF SOME HAMILTONIAN SYSTEMS ON LIE ALGEBRAS

i. Formulation of the Problem of Constructing a Complete Involutive

Collection of Functions

We shall distinguish here the three most important formulations of the problem indicated.

i) The problem of constructing a complete involutive collection of functions on Lie algebras. Let G be a Lie algebra, let G* be the space dual to G, and let K be some class of functions on G* which is closed relative to the Poisson bracket, i.e., K is a Lie subalgebra of C~(G *) (for example, for K it is possible to take all smooth functions on G*, all analytic functions on G*, the space of all polynomials, rational functions, etc.).

It is required to construct a collection of functions fl,...,fs of the class K such that a) {fi, fT}-----Oi 1-~<i, j-.<s (i.e., the subalgebra generated by fl ..... fs is Abelian); b) s = i/2(dimG + indG); c) fl, .... fs are functionally independent almost everywhere on G*.

Such collections of functions are important from various points of view.

a) We obtain a large reserve of regular examples of completely integrable Hamiltonian systems. More precisely, each Hamiltonian vector field v i = sgradfi, i = l,...,s defines a completely integrable Hamiltonian system on each orbit of general position of the coadjoint representation.

b) Collections (fl ..... fs) define (locally) canonical coordinates on each orbit of general position of the coadjoint representation.

c) We obtain a reduction of noncommutative integrability to commutative integrability.

d) We obtain a large reserve of algebraicizable Hamiltonian systems on the space O*_~_R N (see Sec. 2, Chap. 2).

2) The problem of constructing a complete involutive family of functions on a single orbit chosen individually of the coadjoint representation (in particular on singular orbits). It is required to construct a collection of functions fl,...,fs on G* such that the restrictions fl]~ ..... f~l~ of the functions fl ..... fs to the given orbit ~ form a complete involu- tire family of functions on G.

The importance of such collections is that we obtain regular examples of completely integrable Hamiltonain systems on symplectic manifolds.

Problems i and 2 can be strengthened as follows: include in a complete involutive family of functions a given Hamiltonian H which is interesting, for example, from a physical point of view.

3) The problem of constructing a complete involutive family of functions on a given symplectic manifold. Let (M 2n, m) be a symplectic manifold. It is required to construct a collection of functions which almost everywhere are functionally independent and are pairwise in involution.

if the manifold M 2n is homogeneous, then we return to problems i or 2, since any homogeneous symplectic manifold is an orbit of the coadjoint representation.

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methods of constructing functions in involution (4s)

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