Section operators

of rotation of the rigid body. Then the kinetic energy of the fluid and rigid body system has the form T-~2-~(A~j~%+Bue~j)+C~j~j, where Aij, Bij, C ij are constants depending on the shape of the body and on the densities of the body and the fluid; repeated indices are summed from i to 3. Let N = (y~, Y2, Y~), where Yi = 8T/8~i, K = (xl, x=, xa), where x i = 8T/~u i. Then the inertial motion of the rigid body in the ideal fluid is described by the equations

{

@=NXo-I-KXU, dK

(i0)

KX~,

where U = (ul, u2, ua), co = (c01, c02, ~a). The kinetic energy of the rigid body is an arbitrary positive-definite, homogeneous quadratic form in the six variables ui, mi; it is thus determined by the 21 coefficients (Aij, Bij, Cij) = (lij). Equations (i0) in the general case have the three classical Kirchhoff first integrals K 2 = K~ + K~ + K~ = const, NK = KxN x + KyNy KzN z = const,, and, finallyi T = const. THEOREM 3.10.1 (see [99~ i01~ 102]). The system of Euler equations for the Lie algebra E(3) of the Lie group of motions of Euclidean space R a coincides with the equations of inertial motion of a rigid body in an ideal fluid. 3.11. Equations of Inertial Motion of a Rigid Body in an Incompressible, Ideally Conducting Fluid. We consider the classical equations of the magnetohydrodynamics of an incompressible, nonviscous, ideally conducting fluid

I ~ T + (rot ~) X ~-- p- (rot H) • H - - grad II, OH

[~

= rot (~, X H).

As has been shown in [17], these equations have as the simplest finite-dimensional analogue the equations

{M= [~,M]+ lJ,h'],

(11)

on the Lie algebra so(n) of skew-symmetric matrices. Let G be an arbitrary Lie algebra over the field k. We construct a new Lie algebra Q~,~(O), ~, ~6k, containing G as a subalgebra. As a linear space Q=,~(G) is the direct sum G e G . The elements of the first term we denote by a6O, while the elements of the second term we denote by eb, i.e., any element of Q=,o(O) has the form x +ey, x, y6O. On ~=,s(O) we define the product [Xl@TYl, x2@eF2]----[xl,x2] +~[y~, y2]@a([yl, x2[@[Xl, y2]nualyl,yJ). This operation gives the structure of a Lie algebra on~=,~(O). We introduce the notation ~(G) = G + eG, where g2 = 0 for the Lie algebra Q0,0(O). It may be assumed that,Q(O)*-------O*.4-cO~. We denote / s (O)~ by f = x * + e y ~, where X% y*s THEOREM 3.11.1 (see [92, 97]). Let x + e y G e ( O ) , x * + e y * s *. Then ad~+~y(x*@ey*) ~ad]x*+ad~y*+sad~v *, and therefore the Euler equations d=ad~ca )(a), =ca(o) * have the form

X = adxX-r- advY, /~---- ad;Y, where a = (X, Y) and C(a) = (x, y). In the case where G = so(3) the Euler equations for the Lie algebra ~(so(3)) coincide with the finite-dimensi0nal approximations (ii) of the equations of magnetohydrodynamics. 4.

Section Operators

4.1. General Constructions and Definitions. For finite-dimensional Lie algebras there is a simple and natural procedure for constructing dynamical systems which are analogues of the equations of motion of a multidimensional rigid body. These systems are constructed on the basis of the section operators introduced by Fomenko in [105, 106]. Let H be a Lie algebra, let ~ be the corresponding Lie group, let p:H ~ End (V) be a representation of the Lie algebra H in the linear space V, let e:g~-~Aut(V)be the corresponding representation of the group, and let ~(X) be the orbit of the element X % V under the 2698

action of the group $. If we prescribe a linear operator Q:V § H, which we call a section operator, then on the orbits there arises a natural vector field X = p(Q(X))(X). Giving such an operator also makes it possible in some cases to define a sympiectic structure on the orbits (see [102, ii0]). For applications a special class of section operators is of interest which form a multiparameter family whose basic parameters are two elements, namely, aEV and b~K~a, where @a(h)=p(h)(a), h~H. We shall now construct a decomposition of the Lie algebra H into a s ~ of four linear subspaces K + B + R + P and an analogous decomposition of the space V into a sum of four subspaces T + B + R + Z. After this the section operator Q:V + H will be defined as a linear mapping preserving this decomposition into four pieces. In other words, the matrix of this operator will be block-diagonai; four matrices of smaller size occur along the diagonal. Thus, let a be an arbitrary point of general position in the space V, i.e., the orbit of the action of the group $ passing through it has maximal dimension. Let K be the annihilator of the element a in the Lie algebra H, i.e., K = K e r ~ = , where the linear mapping ~=: H + V was defined above. If the point a is of general position, then the dimension of the annihilator K is smallest. Let b~K be an arbitrary element. We consider the action of the operator p(b) on the space V. We denote by M the subspace Ker p(b)~V. Let K' be an arbitrary algebraic complement to the algebra K in the Lie algebra H, i.e., H = K + K' and K n K' = 0. The choice of the plane K' is not unique. The possibility of varying this algebraic complement leads to the appearance of an additional family of parameters in the construction. It is clear that a~A4. By the definition of the plane K' the mapping ~a: H-+V takes the plane K ~ monomorphically into some plane~aK'~V. Since ~aK'=OaH, the plane OaK r does not depend on the choice of K' and is uniquely determined by the choice of the element a and the representation p. We suppose that in the annihilator there exists an element b such that the space V decomposes into a direct sum of two subspaces M and Im p(b). For example, for such an element b it is possible to take semisimple elements of the annihilator K. The plane ~aK' intersects M and Im p(b) along planes which we denote by B and R', respectively. We obtain a decomposition of ~aK' into the direct sum of three planes B § R' + P, where B and R' are uniquely determined, while the additional plane P is not uniquely determined and contributes its collection of parameters. We consider the action of the transformation 9(b) on the plane Im p(b). It is then clear that p(b) isomorphically maps Im p(b) onto itself. In particular, the operator p(b) is invertible on the plane Im p(b). Let p(b) -l be the operator inverse to p(b) on Imp(b). We set R=4p(b)'I(R')~ We then have.p(b) :R-+R'. The plane R is uniquely determined. In Im p(b) we consider the plane Z which is the algebraic complement to the plane R. Then Imp(b)=Z~-R', R'~R. Let T be the complement to the plane B in M (Fig. 6). We finally construct a decomposition of the space V into the direct sum of four planes V = T + B + R + Z. Here the planes R, B, M, Im p(b) are uniquely determined, while the planes Z, T are not unique and contribute their collection of parameters to the construction. If in V there is given a scalar product, then the planes Z, T are uniquely determined as the orthogonal complements~to the corresponding subspaces. Since K ~ is isomorphic to ~aK', it follows that K' = B + R + P, where B = ~ I B , R=~IR, p=o~Ip. Thus, a multiparameter~family of decompositions of four spaces K + B + R + P has been defined.

of the Lie algebra H into a direct sum

We define the section operator Q:V + H, Q:T + B + R + Z + K + B + R + P, setting

Q =

~7'

0

0

0

(i ~

D~/

here D:T + K and D':Z + P are arbitrary linear operators, ~[|:B-+B is defined by the formula ~.a(h)=9(h)(~ and ~1op(b):~-+R. Thus, the operator Q has the form Q(a, b, D, D'). We can now construct the dynamical system X = p(Q(X))(X). The choice of a as a point of general position in V is occasioned by the fact that in this case the dimension of the plane K' is maximal, i.e., the operators ~ o p ( b ) and ~ i have the largest domain. In the case of the coadjoint representation of the Lie algebra the construction of section operators presented above can be detailed as follows. Let G be the Lie algebra, let G* be, as usual, the space dual to G, and let ~EG*. We denote by Bf the bilinear, skew symmetric form (x, p)~+[([x,y]), ~ y~G, defined on the Lie algebra G. For any subset A c G we 2699

K< ImjJ(8}

Z/ wR~

1

P M"Ker2(B~

Fig. 6

l< k

Fig. 7

denote by A f the orthogonal complement to A relative to the form Bf. We recall that an e l e m e n t / 6 G *

is regular in the space G* if d i m G / = i n f d i m G <

If f

is a regular element of the space G*, then the subalgebra G f is commutative. Restricting the representations ad and ad* to H = G f, we obtain on the basis of familiar theorems from the theory of representations of Lie groups and algebras the root decompositions

~0

~0

~6H*

~6H*

of t h e spaces G* and G, r e s p e c t i v e l y . We f i x a c o v e c t o r adx*(a), ~a: G-+G*. I f a~Gz*, b6G,, then ~(b)6G~+,.

aEG* and

c o n s i d e r t h e mapping ~ a ( x ) =

We can now c o n s t r u c t t h e m u l t i p a r a m e t e r f a m i l y of s e c t i o n o p e r a t o r s

C(a, b, D):G*--~G.

Let f~G* be a regular covector. We suppose that the Lie algebra G possesses the following property. In the space G~ there is an element a~O0 *, such that the mapping @=:G,-+G,* is an isomorphism for all roots ~ ~ 0. Let D :G0*-~G0 be an arbitrary self-adjoint linear operator. We set by definition

C=C(a,b,D)--( in correspondence with the above decompositions (see Fig. 7).

D

0 .\ ~;1oad~)

into a direct sum of the spaces G* and G

With the help of the construction indicated of A. T. Fomenko of section operators all section operators presently known for which the Euler equations (2.1.1) corresponding to them are completely integrable are obtained. 4.2. Section Operators on Symmetric Spaces. In the case of symmetric spaces section operators make it possible to construct not only flows but also new symplectic structures. Let ~/$ be a compact symmetric space. Then the Lie algebra G of the group ~ decomposes into a sum of planes H + V where H is the stationary subalgebra and V is the tangent space to | The subalgebra H hereby acts in coadjoint fashion on the plane V, and we can apply the general construction of part 4.1 to this representation. The decomposition V = T + B + R + Z defining the section operators then takes the following form: T is the maximal commutative subspace in the plane V, R = R', Z = 0, ~a K' + T = V = T + B + R. If C:V + H is the section operator, then on the orbits G ( X ) c V there arises an exterior 2-form Fc(X; ~, ~) =, where ~, N~TxO, and <X, Y> is the Cartan-Killing form of the Lie algebra G. It turns out that there exist rich series of symmetric spaces and section operators for which this form defines (almost everywhere) on the orbit a nondegenerate closed 2-form, i.e., a symplectic structure. This structure, generally speaking, is not invariant under the action of the group.

2700

Together with the form FC, on the orbits there is defined a flow X constructed by means of some other section operator Q. A natural question arises: for which operators C is the 2-form Fc.closed on the orbits and nondegenerate? Further, for which operators C and Q is the flow X Hamiltonian relative to the form FC? It turns out that for many symmetric spaces rather complete answers to these questions can be given; see [102, 105, 108, 132]. If a semisimple Lie group ~ is taken as the symmetric space, then it can be represented in the form ~X~/~, where the involution o : ~ X ~ - + ~ X ~ is given by the formula o(x, y) = (y, x). The corresponding decomposition in the Lie algebra G = H + V has the form V = {(X, -X)}, H = {(X, X)}. Here the Lie algebra H and its realization in G are denoted by the same letter. We have oV = -V, oH = H. It is easy to verify that the 2-form F C constructed above here becomes a canonical symplectic structure on the orbits of the adjoint representation for a suitable choice of the section operator; see [102, 105]. As an example of symplectic structures of the form F C we consider symplectic structures on spaces of maximal rank. Proposition 4.2.1 (see [105]). If the space ~ = ~ / ~ has maximal rank, then the 2-form F C on V is generated by the curvature tensor of the symmetric space ~. More precisely, Fc(X; ~, q)=4, where R is the curvature tensor and a~6T is a fixed vector. A complete description of symplectic structures of the form F C can be found in the work [105] (see also [103, 108, ii0]). There are other constructions of symplectic structures connected with the curvature tensor (see, for example, [94, 147]). 4.3. A Complex Semisimple Series of Section Operators. We shall define analogues of the equations of motion of a solid body on an arbitrary semisimple Lie algebra (see [65, 66, iii]). Let G be a complex semisimple Lie algebra, and let G = T + V + + V- be its root decomposition. Let ~, b~T, a=/=b be two arbitrary regular elements of the Cartan subalgebra T. We consider the operator ada:G-+G. It is clear that the operator ad a preserves the root decomposition. Since a is of general position, the operator ad a is invertible on V-~V§ According to the general method of part 4.1, we define a section operator ~a,o,o:G-+O as follows:~a,~,o(X)=~a,b(X~)-~D(t)~adT~ad~X~+D(t), where X = X' + t is the unique decomposition of the vector X according to V and T, and D:T § T is an arbitrary operator symmetric on T relative to the Cartan-Killing form. The operator ~a,~,o is parametrized by a, b, D. In the Weyl basis (E=, E_=,f-f~)of the semisimple Lie algebra G the operator ~ is giuen by the matrix kl

0

0 0

E~

0

E c,

D

Ho~

kq k,

0

"~aeD 0

~,q

where k==~(b)=(~-i and q = dimV • is the number of positive roots. The operators ~ constructed play the role of inertia operators for complex semisimple Lie algebras. 4.4. A Compact and Normal Series of Section Operators. We shall construct a family of Hamiltonian systems on an arbitrary compact real Lie algebra by using for this the real forms of complex simple Lie algebras (see [65]). E a c h complex semisimple Lie algebra G possesses a compact real form G u. We recall that Gu={E=+E_=,Z(E=--E_=),iH ~. Let a, b6iTo (where T O is the real subspace in the Cartan subalgebra T spanned by all roots filET) be elements of general position. It is clear that ada(E~+E_~)~(ar)(i(E~--E_~)), ada(i(E~--F_~))=--~(~O X(E~+E-~), where a-----iaI, a1~T0 . Thus, the operator ad a rotates the vector E = - ~ _ ~ into a vector proportional to i(E~--E_~) and conversely. Therefore, all vectors E = ~ - E _ = and s are eigenvectors with eigenvalues ~(b)~(a)-1-----~(b~)~(ar) -~, a----ia~, b=ib ~, a~,b~@To . We define the operator

2701

~-~=,,,D:Ou-+G u as ~ ( X ) = ~ (X'+t)-~=,b(X')+D(t)=adaladoX'+D (t), where X = X' § t i s the unique decomposition of X into components such that t~iTo, X'!iTo. In the basis {(E=+E_=), i (E= -- E_=), ~/-/=} the operator T is given by the matrix ~1

0

E~z+ E_o~

0

0

"~, ~,

0

i (Eo~--E_oO

0 0

0

where the numbers

%~-c~(b)/~z(a)are

k,q

D

0

i H~

real.

We now construct analogous operators on some simple compact real Lie algebras corresponding to the classical normal compact subalgebras in complex semisimple Lie algebras. In each compact form G u we consider the subalgebra G n, called the normal subalgebra, which is spanned by the vectors E=+E_=, where ~ runs through the set A of all roots of the Lie algebra G relative to the Cartan subalgebra T. Since all these vectors are eigenvectors for the operators @ of the compact series, it follows that on restricting them to the subalgebra G n w e obtain a normal series. These operators simply coincide with CPa,b:Gu--+Gu, cp(X)=ada-ladb(X), X60~, a, bEiTo, o;(a)=/=O,~(b)~=O. In the basis {E~-E_~}the operators ~p are given by matrices

(:0)

We n o t e t h a t h e r e a, bGG~, i . e . , operators of the normal series require for their definition elements of a larger Lie algebra. T h i s d ' i s t i n g u i s h e s t h e n o r m a l s e r i e s f r o m t h e c o m p l e x and compact series for which the elements a, b belong to the Lie algebra studied itself. By t h e way, an a n a l o g o u s phenomenon o c c u r s i n t h e s t u d y o f s e c t i o n o p e r a t o r s on s y m m e t r i c s p a c e s ; see [105]. We s h a l l show t h a t H a m i l t o n i a n s y s t e m s o f t h e n o r m a l s e r i e s c o n t a i n t h e c l a s s i c a l equations of motion of a multidimensional s o l i d body w i t h a f i x e d p o i n t . We c o n s i d e r t h e L i e a l g e b r a s o ( n ) and r e p r e s e n t i t i n t h e f o r m o f a n o r m a l s u b a l g e b r a i n t h e L i e a l g e b r a s u ( n ) . We imbed s u ( n ) i n s t a n d a r d f a s h i o n i n u ( n ) and c o n s i d e r two r e g u l a r e l e m e n t s a , b i n t h e C a r t a n s u b a l g e b r a iT 0 i n u ( n ) [ a n d n o t i n s u ( n ) ] 0 L e t a = d i a g ( i a l . . . . . ia~), b=diag(ib~ . . . . . H~), at, btfiR, at=/=+ap b1=/=bi for l--/=j. Then the operator ~a,o:Gn-+Gn acts as follows: a (b) Therefore,

the Hamiltonian

~X] h a s t h e f o r m

[bq--b~

n

n

q=l

q=l

9

S u p p o s e now t h a t

system X=[X,

a = - - i b 2, i . e . ,

ap=b~.

bi--bq t ai--=q

]

Hence H

1

"~ti"~-~aXiqXqi('~j+aq

I

a,+aq

).

q--1

Thus, for a = - - i b 2 we obtain the system of equations already known to us from part 2.2 of the dynamics of a solid body with a fixed point. Moreover, the operators ~a,b of the normal series contain the classical operator ~X = IX + XI, where I is a real diagonal matrix. 4.5. Section Operators for the Lie_Algebra of the Group of Motions of Euclidean Space. The Lie algebra E(n) of the group of motions of Euclidean space is the semidirect sum so(rD@R n, where ~:so(~)-+End(R n) is the differential of the standard representation of the group SO(n) 2702

in the s p a c e R ~, and R ~ is considered as a commutative Lie algebra. The space E(n)* dual to E(n) we identify with E(n); for this we define a nondegenerate scalar product in E(n) (noninvariant). We ha~e E(n)* as a linear space. Let E(n) =so(n) + R n be the Cartan-Killing form of the Lie algebra so(n), and let (X, Y)e be the Euclidean scalar product on R a. We then set ((x~, y~), (x~, 9~))=B(x~, x2)+ (y~, y2)s, x~, x ~so(n)~y~,y~6R n. We now represent all subspaces in E(n)* as planes in E(n), using the identification of E(n)* and E(n) described above. The orthogonal complement to the subspace W in E(n) or in E(n)* relative to the scalar product B(X, Y) + (Z, R) e we denote by W • Let a be of general position~ and letK~---Ker~a. Then Oa maps K • into K *• where K = K* under the identification of E(n)* with E(n). The mapping ~ a : K i - ~ • is an isomorphism. According to the general method, let E(n)=K• E(n)*=K*• * and let a6K*, b~K, where a x6K . ys , then C (a, b, D) (z)~-~ad~ (x)+D (y), is of general position. If z-~-x+y~E(n)*, *• * where D : K * - ~ K is an arbitrary self-adjoint linear operator. We note that in the case of E(3) the Euler equations on E(3) * for the kinetic energy

T=2=IX(C(~, b, D)(X)) can be integrated explicitly, since they have the form

x~ = O, 9

where

a~

6

~,

, 2 c)T OT ~=(b2a2-r2b~aO/b 2, x~---a~, Y~=~d~"

4.6. Section Operators for Lie Algebras of Type ~(G). For Lie algebras ~(G), where G is a semisimple Lie algebra, section operators of three types can be constructed: complex, compact~ and normal operators; see [96, 97]. Proposition 3.6.1. Let G be a complex, semisimple Lie algebra, and let aE,Q(G)* be an element of general position; i.e., the orbit of the coadjoint representation of the Lie group ~](~) passing through a has maximal dimension. Then K e r ~ = H + s H , where H is the Cartan subalgebra in G. Let b=ba+eb2~KerOa=H+eH~(G),where b I is of general position. Then Ker0(b)= H+sh'~P~(O)*~--O+eO. It is asserted that@a(Ker~a• and p(b)(Kerp(b)• • for any b6Ker(D~, where b• denotes the orthogonal complement relative to the direct sum of the CartanKilling forms on G. We define the "complex" series of operators C:~(G)* ~ fi(G) by the matrix

C - - C (a, b, D ) =

D

i n c o r r e s p o n d e n c e w i t h t h e decomposition~(G)=Ker@a+Ker(I)a• * =Kerp(b)+Kerp(b)-L; h e r e D:Ker:p(b)-+KerOa i s an a r b i t r a r y s e l f - a d j e i n t l i n e a r o p e r a t o r . The " c o m p a c t " s e r i e s o f s e c t i o n o p e r a t o r s ~C : f~ (Gu)*--~Q (Gu) i s a l s o g i v e n by f o r m u l a ( 1 ) , o n l y D:Ho+eHo--~Ho+eHo(H o i s t h e r e a l p a r t o f t h e C a r t a n s u b a l g e b r a H i n G). The " n o r m a l " s e r i e s o f s e c t i o n o p e r a t o r s C : ~ ( G ~ ) * - - ~ ( G n ) i s the restriction t o ~(G n)

of operators of the "complex" series. The kinetic energy T=<X,C(a,b,D)(X)> structed has the form

on ~(so(3))* for the operator C(a, b, D) con-

T =- k ~x~+ ks (y~ + y~) i-, -kiy~-~-2ks (x~ya + x2Yi) -4:-(ks -+-/~4)X3y3" We r e m a r k t h a t t h e k i n e t i c e n e r g y c o n s t r u c t e d on ~ ( s o ( 3 ) ) * i s t h e l i m i t o f t h e k i n e t i c e n e r g y of the classical first Clebsch case of inertial m o t i o n i n an i d e a l f l u i d when one o f i t s

parameters tends to zero. 4.7. Section Operators for the Lie Algebras su(n)~C ~ and u.(n)OC.n._ These constructions were carried out by A. V. Bolsinov. Let G(rZ)=su(rt)@C n, F (n)=u(~)@C ~. We identify the spaces

2703

F(n) and G(n) with the dual spaces by means of the scalar product< (Xi, Z 0' (X~, Zs)> -~tr X~X~-~ 2Re(Z~, Z~), where ( , ) is the Hermit• scalar product in the space C n. In the case of the Lie algebra G(n) we label the operator ad* with a prime: ad*' = ad*(G(n)). We consider the subspaces LcF*(rt) and i ' < O * (n):

"'ixn-'O x

x~. . . . . Xn-~, x~R, ~ x~ ~ 0

and

L--

*x"-~O x

)

...~ Here we use matrix representations of the Lie algebras:

6-TZ--ff~

t\~-Z...--ff1~-/~

We choose regular elements a6L, a'6L', such that x1~A0 ..... Xn_1~0~ x ~ 0 , xi--xj~-O for all i, j. We call the set Ann(a)--{b@F(~)lad~(a)=0 } the annihilator of the element a@L. Similarly, Ann (a')={b'~G (~)[aN,, b,(af )=0}, Ann (L)={bEF (~)Ia db* (x)=0 for all xCL}, Ann (L')= {b'CG (n)]ad~',(x)=0 for all xEL'}. LEMMA 4.7.1.

The following equalities hold: a) Ann(a)~-Ann(L)=%;

b) Ann(a')=Ann(L')=L '.

Let M and M' be the algebraic complements to L and L', i.e., F ( ~ ) = M | 1 6 3 Q(~)=/,'I'@Lq We consider the mappings qDa:M -+F* (n) and~a,:s defined by the formulas CDa(X)=adxa and ~a' (x)----adx (a'). LEMMA 4.7.2. a) We have(Da(/Fl)=L "i and ~Da:M-+L i is an isomorphism, L '• and ~a,:M'_+L'• is an isomorphism.

b) We have ~=,(7Plr)=

Let bCL, b'6L ~ be arbitrary elements. LEMMA 4.7.3.

The following inclusions hold: a)ad~(L~)czL•

ad~',(L'i)cL'•

By the general procedure expounded above we can now construct multiparameter families of section operators C : F * ( r t ) - + F ( r ~ ) a n d C':G*(~)-~O(~). Definition 4.7.1. Let ~6L, b~L, where a is an element of general position. Let z = x ~ yCL. We then set by definition C(a, b, D)(z) = ~ a -iadbj(x)@D(y), ~, where D:L + L is an arbitrary linear operator.

yEF*(~), xGL•

Definition 4.7.2. Let a'6_L', b'fiL', where a' is an element of general position. Let z = x-7'9EO*(tz), x6/'1, y~t'. We then set by definitionC'(~', b', D')(z)=~a, lad*~,(x) ~cD' (y),where D': L' + L' is an arbitrary linear operator. Let ~t:L~L' be the orthogonal projection, and let j~:L'~L be the imbedding. Then to each section operator C'(a ~, b', D'):O*(IZ)-+O (N) we can assign an operator C (a', b', D'):F* (~)-+F (t~), where D'--j~oD'o~ (the elements a', b' are considered as vectors in L). The connection of the operators C' (a', b', D') and C (~', b', D') is given by LEMMA 4.7.4.

F*(rt)=G*(n)| 5.

Let ~:F*(r~)-+G*(~) be the projection corresponding to the decomposition Then C(a', b', D')=joC'(a', b', D')o~.

~ and let j:G(rt)-+F(r0 be the imbedding.

The Bi-Hamiltonian Property of Euler Equations and Symmetric

Lie Algebras 5.1. A Characteristic Property of Inertia Tensors. Before formulating the main resuit - a theorem on the characterization of quadratic functionsF(x)=2-1<x,~a,b,D(x)> - we introduce in the ring C~(G) of smooth functions on the Lie algebra G an additional Poisson

2704