Euler equations on solvable Lie algebras

THEOREM 2.4.2 (A. V. Brailov). Let (G0,ffR)be a split simple Lie algebra; let G o = K + L be the Cartan decomposition indicated above; letias be an arbitrary element; let G~ be the centralizer of a in Go; let Cent G~ be the center of the Lie algebra G~; let the element bGLnCent O$; let D:Ka-+K a be a symmetric operator invariant relative to all inner automorphisms of the Lie algebra Ka; let Iz,...,l r be collection of homogeneous, algebraically independent invariants of the Lie algebra G O of degrees m I + l,...,m r + I, where ml,...,m r are the exponents of the Weyl group. Then for the operaterlqa,~,D, defined by the matrix ~ a , b, D --~i

relative to the decomposition A"~--[a, LI.-}-ffa , the Euler equation A~[X, ~a,b,D(X)], XEK, has the following integrals: the integrals I{,a(X), which are by definition the functional coefficients of ~2 in the polynomialP~,a(%, X)~-f~(X@%a), and the integrals[g(X)= (g, X>, where gGK a. All integrals of the formll[,w commute pairwise with each integral s From the integrals I{.~ and i it is possible to select a number of independent integrals equal to q + dimK a, where q is the number of integrals selected here of the form l{,a. For the number q we have the following expression: q = i/2(dimK/Ka + r k K - rkKa). The rank of the reductive Lie algebra K is equal to the number of odd exponents of the series ml,...,m r. Remark. In a manner similar to the way in which Theorem 2.4ol augments Theorem 2.1.1 of A. S. Mishchenko and A. T. Fomenko on the independence of integrals of the so-called "compact" series, Theorem 2.4.2 augments Theorem 2.1.1 on the independence of integrals of the "normal" series for the case of singular operators ~a,b,D3.

Euler Equations on Solvable Lie Algebras 3.1.

Euler Equations on Borel Subalgebras of Semisimple Lie Algebras.

Let G be a c o m -

plex simple Lie algebra, let H be its Cartan subalgebra, let G-----f-f@~G=be the Caftan de~0

composition, and let {hi, e~} be the Chevallier basi~.

We consider the Borel subalgebra

in

~>0

to it there corresponds the Lie group ~G. In the Weyl group W(G, H) of the Lie algebra G there exists an element w 0 of greatest length. A complete involutive family of functions on Borel subalgebras BG in simple Lie algebras was constructed in the works of Trofimov [88, 90]. THEOREM 3.1..1 (see [4]). Let d i be semiinvariants of the representation Ad* of the Lie g r o u p ~An. There exists an open, dense subset U c B A ~ * such that if a function f on BA~ depends in polynomial fashion on the functions ~i(X~-~a),i= 1..... n,%6R, a6U, then the system of equations x = {x,dfx} is completely integrable in the Liouville sense on orbits of general position of the representation Ad* of the Lie group ~Anl An analogous construction can be carried out for ~Sp(n)(see [88]). THEOREM 3.1.2 (see [88]). Let d i be semiinvariants of the representation Ad* of the Lie group ~Sp(n). There exists an open, dense subset~UcBSp(n) * such that if a function f on BSp(n)* depends functionally on the functions di(x+%a), i=l,..., n, %ER, a~U, then the system of equations x = {x, d[x} is completely integrable in the Liouville sense on orbits of general position of the representation Ad* of the group!~Sp(n). An explicit description of the semiinvariants for'~An can be found in the work of Arkhangel'skii [4] and for ~Sp(n) in the work of Trofimov [88]. In the work [90] a description is given of the semiinvariants and, in particular, the invariants for all Borel subalgebras BG in simple Lie algebras G. Let A~(X) be the lower-left-corner minor of order i of the matrix X, and let Oij(s) be the border of the minor As(X) corresponding to the element xij. THEOREM 3.1.3 (see [88]). Suppose a function f on BSO(n) ~ depends functionally on the semiinvariants of the representation Ad* of the Lie group ~SO(n) and, further, in the case n = 2k on the coordinates of the maximal Abelian subalgebra of BSO(n), i.e., on Yi,j+n (i + j < n), in the case BSO(4k + l) on the shifts Ak+l and the coordinates Yi,j+k (i + j < k) of

2722

the Abelian subalgebra, and in the case BSO(4s + 3) on the shifts Ok+1,~+~(k--l) and the coordinates Yi,j+k (i + j < k) of the Abelian subalgebra [SO(N)is realized by matrices which are skew-symmetric with respect to the secondary diagonal]. Then the s y s t e m o f equations i = {x, dfx} is completely integrable in the Liouville sense on orbits of general position of the representation Ad* of the Lie group ~SO(n). The method of chains of subalgebras makes it possible to construct a complete, family of functions also on BG for special Lie algebras.

involutive

THEOREM 3.1.4 (see [88]). If a function f on BG~ depends functionally on the semiinvariants of the coadjoint representation Ad* of the Lie group ~G2 and on the coordinates of the maximal Abelian subalgebra in BG 2, then the system of equations i = {x, dfx} is completely integrable in the Liouville sense on orbits of general position of the representation Ad~o,. The generalized shift method of the work [90] can be applied to construct a complete, involutive family of functions on the space BF~, BE~. THEOREM 3.1.5 (see [90]). In the space of polynomials on BF~ and BE~ a finite-dimensional subspace can be explicitly produced which is invariant with respect to the representation Ad* of the Lie group ~F4 or ~E6 and such that if a function f depends functionally on the shifts of the basis functions of the space W, then the system of equations i = {x, dfx} is completely integrable in the Liouville sense on orbits of general position of the representation Ad* of the Lie group ~F4 or ~E6. To construct a complete, involutive family of functions on BG it is necessary to know the index of the Lie algebra BG. This problem is solved by the following theorem due tO v. V. Trofimov. THEOREM 3.1.6~(see [90])~ Let G be a simple Lie algebra, let BG be the real form of the Borel subalgebra in G ffescribed above, and let w0Elg/(G, ff) be an element of the Weyl group of maximal" length. If ~ is an orbit of maxlmaI dimension of the representatlon Ad~o, then 1

codim~----~cardA , where A = { ~ @ A l ( - - w 0 ) a ~ s A ~ } , gebra G, and cardS

5 is the system of simple roots of the Lie al-

is the power-of the set S.

In the work [89] section operators of "rigid-body" type are defined for Borel subalgebras BG. Let G be a semisimple, complex Lie algebra and let H be its Cartan subalgebra; we define operators ~a.o:BG*-+BG by the equality ~a,b(x)=ad~adbo(x), where xEBG*, aEH, and for any root ~ of the algebra G relative to H we have a(a)@O, b@BG, and o is the involution of the compact form in G; here BG* is identified with the subalgebra @ R h ~ @ ~ R e = . then

If xCBG ,

x-~h+~x=ea, hEH, where e~ is the root vector corresponding to the root ~.

We define

~>0

the height ht x of an element x: the height of x is equal to the height of a, where a is the minimal root for which x~ ~ 0. THEOREM 3.1.7 (see [89]).

Let G be a simple Lie algebra of type An, Cn, D n or G 2.

b~BG is an element such that the inequalities i ) h t b ~ ] the case Cn, 3 ) ; h t b ~ n

in the case Dn, 4) h t b ~ 3

in the case An, 2 ) h t b ~ n

in the case G 2 are satisfied,

If in

then the Euler

equations x={X,~a,b(X)} on BG* with the "rigid-body" operators ~a,~ are completely integrable in the classical Liouville sense on all orbits of general position of the representation Ad* of the Lie g r o u p ~ G . THEOREM 3.1.8 (see [89]). Let G he a simple Lie algebra of type An, C n, D n or G2, and let O@BO be an element as in Theorem 3.1.7; then a Hamiltonian flow on T*~O, corresponding to quadratic forms with operators of "solid-body" type ~a,~,is completely integrable in the classical Liouville sense. 3.2. Euler Equations on Some Suba!gebras in BG. Chains of subalgebras can be used to construct complete, involutive collections of functions on semidirect s~m~s of Lie algebras. If G = G o + H, then G~ and H* are subspaces in G*. THEOREM 3.2.1 (see [50]). Let G be a Lie algebra which decomposes into a direct sum of an ideal G o and an Abelian subalgebra H: G = G o + He Let F~,...,F m be a complete, involutive collection of functions on G~. Then the collection FI .... ,Fm on G* is complete and commutative, where F~(f)=F~(fo+h)----F~(fo), i = l , 2 . . . . . m , f~G*, f----fo+& fo~Go*, h ~ H * . 2723

THEOREM 3.2.2 (see [50]). Suppose the Lie algebra G decomposes into a direct sum of an ideal G O and an Abelian subalgebra H: G = Go + H. Let I~, 12 be a complete collection of invariants of G relative to the coadjoint representation A d ~ ; let FI,...,F m be a complete, involutive collection of functions on G~. Then FI, F2,...,Fm, Ii, I2,...,I r form a complete, involutive collection of functions for G, where Fz, F2,...,F m is the natural extension of the functions FI, F2, .... Fm to G*. As an application of these constructions due to Le Ngok T'euen; see [50]. THEOREM 3.2.3.

Let L be any subalgebra in the Lie algebra T n of all upper triangular

formL==V~-

matrices of the

it is possible to obtain the following result

RE~, where Eij is the elementary matrix of order n • n and 1
V is an arbitrary subspace of the n-dimensional space of diagonal matrices. Then on the space L* there exists a complete, involutive collection consisting of polynomials which can be produced in explicit form. A general theorem on the existence of a complete, involutive family of functions for nilpotent Lie algebras is proved in the work of Vergne [168]. However, an explicit construction of such a collection of functions is of interest. As an example of this type of theorem we present the following result due to T. A. Pevtsova. THEOREM 3.2.4. Let fi be semiinvariants of the representation Ad* of the Lie group ~n of upper triangular matrices of n-th order with ones on the diagonal. There exists an open, dense subset U c p~ such that if a function f on P~ depends functionally on functions of the form fi(x-~-%a), ZER, a6U, then the system of equations x=ad]r(x)(X ) is completely integrable in the Liouville sense on orbits of general position of the representation Ad~, (Pn is the Lie algebra of upper triangular matrices of n-th order with zeros on the diagonal). For other examples of this type, see [i00]. 3.3. Toda Chains. The systems described here were found in the work of Bogoyavlenskii [123]. Let q = (ql ..... qn) and p = (Pl .... ,Pn) be the coordinate and momentum vectors in n-dimensional Euclidean space. Let ~i, ~2,...,~s be linearly independent vectors in this space,

and let

q~=(aj, q)=~j~q~__ be

linear functions

(t._
~= 1

Toda chains

are

described

by a Hamiltonian

H

=

of the

form

p2_k U (ql, q2 . . . . . q~)'

~ ]=1

n

where U (ql . . . . .

q.) = ~

g~2 exp (2q%).

If the system of vectors {~i,...,~s is arbitrary, then almost nothing can be said regarding the corresponding Hamiltonian system. However, if {~i} form the system of roots of a simple Lie algebra, then it is possible to completely investigate the corresponding Hamiltonian system. We recall that all systems of roots have been classified: there exist 4 series An, Bn, C n, D n and 5 special systems G2, F4, E6, E 7, and E 8. We shall present a description of the corresponding Toda chains. Type An-l. The system of simple roots is a~=el--e2,...,an_1=en_1--~. simple roots is o. . . . . . . . o. The Hamiltonian is

H=89

The scheme of

+... +

1=1 T y p e Bn. The system of simple roots is o--o--...~o~o.

simple roots is a~=et--e~ ..... The H a m i l t o n i a n is H=~

1

~G.

The scheme of

n

~a Pi~Weq'-q~

]=1

. . . _Leqn-l--qn~ , , - -~qn.

T y p e Cn. The system of simple roots is a~=e~--e2 ..... of simple roots is o--o. . . . . . ~o. The Hamiltonian is

2724

an-t=e~-~--~,

~-l~en-1--~,

~2~.

The scheme

n

7=I I

The s y s t e m o f s i m p l e r o o t s i s a~----el--e2, a2-----e2--eal ...1 an-l-~en-l--enl an=gn-l'ren. /~ The scheme o f s i m p l e r o o t s i s Q--o. . . . . ~ The H a m i l t o n i a n i s

• P12"t- eq'--q~ + .. 9@ 8qn-l-qn "~ egn-l+qn"

H-~-'~I

j=l

We note that for the system D, there is an equivalent but more symmetric expression for the Hamiltonian: 1

4

I

-6-(q,--q~--qs--q~)

1t =-f ~ p~ q-eq,.q-eq,.-keq~-ke ~ Type G2._ The s y s t e m o f s i m p l e r o o t s roots is O (~-- 9

is

~=e~--e2,

a2=--2e~q-e2+ea. The scheme o f s i m p l e

The Hamiltonian is 3

H=89~ p]q-e2 qt--qz + e

--2aiq-q,-I-q~.

.

7=I ]

Type F4.

The system of simple roots is ~1=e1--e2,

The scheme of simple roots is o--o~....

~2=e2--ea, ~a-----e3,~4=~-(e4--el--e2--ea).

The Hamiltonian is

4

1

H ~ - ~ | ~,~ p]2 L_,aqt--q2j.l_aq2--q3_l_aq3~.eT, i ~ ~ l ~ i

Type E6. ~4=e3--e4, is

The s y s t e m of s i m p l e r o o t s

c~s=e~--es,

aa-----.--(e~+e~).

1

i s a l = ~ (--e~q-e~q-...qTeT--es),

The scheme o f s i m p l e r o o t s

8

~2=et--e2,

is ............ o!

aa=e2--ea,

The H a m i l t o n i a n

1

2 @ eq~-q, _{_eq~-q~ .@ ~qs-q, -I- eqa-q" "-Fe-(q~+q~) -F.e ' v(-qx+q~+"" +q'-q') ~

1 j=l

a;=--(e~+e~).

I a~=g(e:--es-l-e~+...+e7), a~=e~--e~ . . . . . a~=e~--e~,

The s y s t e m o f s i m p l e r o o t s

is

The scheme o f s i m p l e r o o t s

is ............

The H a m i l t o n i a n i s

o 8

H____E Z p ~ t._er

. . . .q_eq~_q,@e_(q~+r

_~_el-(-q~+q~+..+q,-q~)

7=1

Type Es.

The system of simple roots is ~ i = ~ (--el--es-l-e2d-,..@e7),a2=el--e2 .....aT=e~--

eT, 6~8=--(e1@e2) 9

The scheme of~ simple roots is ..............

The Hamiltonian is

o 8

H = ~1 ~

1

~J--n~-4-eq~-q~4---"" @eq,-q,@e-(ql+q~) @e Y(-ql+q~+''l+q'-r

1=1

All these systems admit a realization on Borel subalgebras in the corresponding simple Lie algebras in the sense of the definition of Sec. 2, Chap. 2. We shall describe the realization of the classical Toda chain of type A n in BA n . We consider the mapping f:R2n-+A, where (Pl 0

Sl p~ 0

0

.n o/l Sn-1

n

is the subspace in the space of lower triangular matrices T~ which is identified with the dual space to the upper triangular matrices. The space A consists of orbits of the representation Ad*. We Set by definition Pi = Yi, sj = exp (xj - xj+1). Under this mapping the 2725

(o:)

Poisson bracket on T~ goes over into the standard Poisson bracket on the

space

If Ht(F)=2tr(F+a) ~-, where F 6 A and

(y~)|

0

a-----

0

9

~

~Tn.

,~-1

then direct computation shows that H=H[ [. This realization makes it possible to construct a complete collection of first integrals of the Toda chain [83]. 4.

Euler Equations on Nonsolvable Lie Algebras with Nontrivial Radical

4.1. Cases of Complete Integrability of the Equations of Inertial Motion of a Multi-dimensional Solid Body in an Ideal Fluid. The system indicated in the title we have already imbedded in the Lie algebra E(n) of the group of motions of the space R n. It turns out that in this case the method of shift of argument makes it possible to construct a complete commutative coilection of integrals on orbits of general position. THEOREM 4.1.i (see [i01, 102]). a) The system of differential equations)~=adQ(x)(X), where Q = Q(a, b, D) is the section operator constructed earlier for E(n), is completely integrable on orbits of general position, b) Let f be an invariant function on E(n)*. Then the functions h~(X)=[(X-b%a) are integrals of the motion for any numbers ~. Any two integrals h A and g~ are in involution on all orbits of the representation Ad* of the Lie group ~(n), and the number of independent integrals of the form indicated is equal to half the dimension of an orbit of general position. Moreover, if ~ is an orbit of maximal dimension +l of the coadjoint representation, then codim~7= t[ n~-]. The integrals are described in the work of Trofimov and Fomenko [I01] or see, for example, the survey [103] or [107] (see also [108, 109, IIi, 130-132]). Complete integrability of the corresponding geodesic flow on the Lie group ~(n)is proved in [95]. The complete commutative collection of functions on the space E(n)* constructed in Theorem 4.1.i plays the role of the "compact" series of integrals for semisimple Lie algebras (see part 2.1 of the present chapter). An analogue of the "normal" series was constructed by A. V. Brailov. Let E(n) be the Lie algebra of the Lie group ~(I~) of motions of Euclidean space R ~. The standard basis of the Lie algebra E(n) consists of elements xij and yk, where xij is an infinitesimal rotation in the (i, j)-th plane, and Yk is the infinitesimal shift in the k-th coordinate, i < j = ]...... n, k = I,...,n. The linear coordinate functions on E(n)* corresponding to the elements xij, Yk we denote by xij and Yk" Let a I ..... a n be arbitrary numbers. We define matrices of dimension(n-~-i)xin+l):E~s=]I6ik6~]]l; here ~ik is the Kronecker n+l

symbol;

n--I

,

Y._---Eu--(n-.cl)-1~E/j;

Y~Eo+Eii(i.~j);

Al~a~Fu, i=l

7=I

xt---HxiiII, where x~i=--xT~,

+ A_~=anE~,.+~, A=A~-FA_t;

x~.n+a=xn+1, k = 0 for k = l .....~ - I ;

X-I---- YIE~.~+~; X=X~n-~' ~-i. i=I

Thus, X, X I, X_ I are matrix functions on the space E(n)*. THEOREM 4.1.2 (A. V. Brailov). Let at> ... >an-t, ~ln~AO, bl, b2..... bn be some numbers. the d~fferential equations i = ad~(x)x, xCE(n)* with the quadratic function n--2 n--1

n--I

2 H = ~'~, "~ b~--bi, x~i__ 2 ~--_lj=~+la~-aJ .=

Then

n--1

yiXin--~L

yi-i'~n Yn

are completely integrable in the Liouville sense on an orbit ~ of the representation Ad* of the group ~(n) in E(n)* for an orbit 0' of general position in E(n)*, A complete collection of commuting integrals is formed by the functional coefficients hk,s (k = 2,...,n + I; s = 0 ..... k) of Xs~-2 in the polynomial hk(~,~-!)=A~(XI~%A~t2(X_~-[-~A_~)), where 5k is the sum of all syrmmetric minors of k-th order. 4.2. Cases of Complete Integrability of th_e Equation of Inertial Motion of a Multidimensional Rigid Body in an Incompressible, Ideally Conducting Fluid. The method of tensor extensions makes it possible to construct complete, com~nutative collections of first integrals for finite-dimensional analogues of the equation of magnetohydrodynamics described above. Let G be a complex, semisimple Lie algebra, and let ~a(G) be the set of functions on G* which are shifts of the invariants F of the coadjoint representation Ad* of the Lie algebra G,

2726