CC BY
8
Владикавказский математический журнал 2011, Том 13, Выпуск 1, С. 59-70
УДК 512.81
ON CONSERVATION LAWS IN AFFINE TODA SYSTEMS1
M. S. Nirova
With the help of certain matrix decomposition and projectors of special forms we show that non-Abelian
Toda systems associated with loop groups possess infinite sets of conserved quantities following from
essentially different conservation laws.
Key words: non-Abelian Toda systems, loop groups, symmetries and conservation laws.
1. Introduction
Classical integrable systems are understood as nonlinear differential equations which can be integrated in one or another sense. According to Liouville [1], the integrability of a system with 2n degrees of freedom is related to the existence of an Abelian n-torus. However, it is not always possible to reveal such tori explicitly in the phase space. For some complicated systems, for example, one can construct general solutions explicitly in quadratures, but with no way obvious to find respective "action-angle" variables. Or one can find certain number of integrals of motion, required by the Liouville-Arnold's theorem, but with the question of involutivity of such quantities still to be answered. In any case, the integrability of a nonlinear system is based on its symmetries. The most popular viewpoint at present is that a system is regarded as integrable, if for the equations describing it one can propose a constructive way to find their solutions, and one can prove that it possesses sufficient number of conserved quantities.
Of special interest are two-dimensional systems which can be represented in the form of the zero curvature condition. These are, for example, the Liouville equation, the Nonlinear Schrodinger equation and its modifications, the KdV equation, the sine-Gordon equation that, also describing, as physicists believe, a system equivalent to the Thirring model, and others [2, 3]. In particular, all these systems are very attractive for investigations in mathematical physics and differential geometry. Here we consider Toda systems associated with loop groups, which actually cover most interesting examples of completely integrable systems [4]. In the case of Toda systems associated with finite-dimensional Lie groups one obtains simple conservation laws whose densities give rise to the so-called W-algebras [5].
It is usually claimed that the complete integrability of such systems should be due to the existence of infinite sets of conserved quantities produced by some current conservation laws. This observation has not received a form of a proved statement yet, and so, it still attracts much attention in the scope of the theory of integrable systems. Besides, it is believed that the classification of integrable systems can be performed along the lines of a "symmetry approach", which is also using the conservation laws [6, 7].
© 2011 Nirova M. S.
1 Partially supported by a grant from Russian Foundation for Basic Research, project № 08-01-00009.
Hence, the main purpose of our paper is to present, in justifiable detail, current conservation laws for non-Abelian Toda systems, which should produce infinite sets of conserved quantities responsible for the integrability of the nonlinear equations under consideration. In particular, our consideration gives a generalization of what was presented in [8]. Our approach, based upon a detailed consideration of the matrices c± entering the equation under consideration, allows one to separate explicitly proper current conservation laws from the simpler WZNW-type conservation laws and those special relations leading to nonlinear W-algebra extensions of Virasoro algebras.
Let G be a complex or real Lie group and g its Lie algebra. Consider a trivial principal fiber bundle M x G ^ M with G as the structure group. Here M is a two-dimensional base manifold, being as usual either c or r2, where standard coordinates z- and z+ are introduced. The Toda equations can be obtained from the zero-curvature condition for a connection on M x G ^ M imposing grading and gauge-fixing constraints on elements of g. In this, the connection is identified with a g-valued 1-form on M, and as such, can be decomposed over basis 1-forms,
where the components u-, u+ are g-valued functions on M. One says that the connection u is flat, and so, the corresponding curvature is zero, if and only if, in terms of the components, one has
The partial derivatives d- and d+ are taken over the standard coordinates z- and z+, respectively.
We study models based on the loop Lie group G = l(GLn(c)) given by the smooth mappings from the circle s1 to the Lie group GLn(c) with the group composition defined point-wise. Its Lie algebra g is the loop algebra l(gln(c)) formed by the smooth mappings from s1 to gln(c). The circle s1 is parameterized here by the set of complex numbers A of modulus 1, and extending the unit circle to the whole Riemann sphere we introduce the so-called spectral parameter denoted here by the same symbol A. Here we consider a non-Abelian analogue of the standard grading [9, 10] and use a representation where the gradation is over powers of A.
To obtain nontrivial systems from the zero-curvature condition one should impose on u grading and gauge-fixing conditions [4, 11]. We assume that g is endowed with a z-gradation,
and for some positive integer l the subspaces g-m and g+m for 0 < m < l are trivial. Note that go is a Lie subalgebra of g. Denote by Go the connected Lie subgroup of G which has go as its Lie algebra. It can be shown that the connection u can be brought to a form given by the components2
2. Toda systems and their simplest symmetries
u = u_dz + u+dz+,
d_u+ — d+u_ + [u_, u+] = 0.
(2.1)
u_(A) = y 1 d_Y + A 1c_, u+ = A7 1c+y,
(2.2)
2 Hereafter we put I = 1, that can be done without any loss of generality.
where 7 is a mapping from M to Go, and c_ and c+ are some fixed mappings from M to g_i and g+i, respectively, such that d+c_ = 0, d_c+ = 0. The zero-curvature condition for the connection with the components (2.2) produces the Toda system that is the nonlinear matrix differential equation
d+(7-1d_ y ) = [c_ ,7-1c+7j. (2.3)
The Toda equation can also be written in another equivalent form,
d- (d+YY = [yc- y 1,c+],
in which case the connection components are
u- = A 1yc-y 1, = —d+YY 1 + ac+.
(2.4)
(2.5)
When the Lie group Go is Abelian the corresponding Toda system is said to be Abelian, otherwise one deals with a non-Abelian Toda system [11-13]. The complete list of the Toda systems associated with finite-dimensional Lie groups is presented in [14]. For the case of loop Lie groups the respective classification was performed in [15, 16].
Let n- and n+ be some mappings from M to G0 subject to the conditions
= 0, = 0.
If a mapping y satisfies the Toda equation (2.3) then the mapping
Y'= n-1Yn-
satisfies the Toda equation (2.3) with the functions c_ and c+ replaced by
/ -1 c- = c_n-,
c+
n_lc+n+.
(2.6)
(2.7)
In this sense the Toda equations defined with the fixed functions c± and c± related by (2.6), (2.7) are equivalent. It is clear that conservation laws when established in terms of such transformed quantities should be the same as they would be for the original ones.
If the mappings n_ and n+ satisfy the relations n_1c_n_ = c_, n^ c+n+ = c+, then the mapping 7 satisfies the same Toda equation as the mapping 7. Hence, in such a case the transformation described by (2.6) is a symmetry transformation for the Toda equations. It gives simplest symmetries of the Toda equations, inherited from the WZNW theory [5].
Now, the mapping 7 takes values in the Lie group of complex non-degenerate block diagonal n x n matrices, possessing the partition just according to the Z-gradation under consideration, that is
/ r 0 ■ ■ ■ 0 \
0 r2 ••• 0 7 = .....
\ 0 0 ••• rr /
with ra taking values in the space of complex non-degenerate ka x ka matrices. The fixed matrix-valued mappings c_ and c+ are explicitly of the forms
c_ =
0 0 ••• 0 C-r 0 c+1 0 ■ ■ 0 \
C-i 0 ••• 0 0 0 0 C+2 ■ ■ 0
0 C-2 ••• 0 0 > c+ =
0 0 0 ■ ■ C+(r-l)
V 0
C-(r-l) 0 /
V C
+r
00
0
0
where C-a denotes a ka+i x ka submatrix, and the block C+a means a ka x ka+i submatrix. It is worthwhile noting that the following relations hold:
s-1Sc-S-1 = c-, sSc+S-1 = c+, (2.8)
where S is a constant diagonal n x n matrix
/ slki 0 0
S =
00
k2
0 0
(2.9)
Ikr /
with s being the rth principal root of unity, s = e2ni/r, so that in terms of the block submatrices
Sa,b = S^fca dab-
The matrix S satisfies the relation
Sr = In,
where In is the unit n x n matrix.
In terms of the submatrices the n x n matrix Toda equation (2.3) takes the form
(r-1 d_ra) = C_(a_i)r-_11C+(a_l)ra - ^ C_a, (2.10)
with a = 1,2,..., r,... and the periodicity condition imposed as follows:
ra+r = ra > C_(a+r) = C_a; C+(a+r) = C+a -
These submatrices, if transformed according to (2.6), (2.7), would look here as follows:
ra = n_a ran_a, C_a = n_(1a+1) C_a^_a, a = n_1C+an+(a+1), with the block diagonal matrices n± defined by (n±)ab = n±adab.
3. The mappings c_ and c+ as linear operators
Denote by k* the minimum value of the partition numbers {ka} and suppose that the fixed mappings c_ and c+ are chosen to be constant. Besides, we assume that the submatrices C_a and C+a are of maximum ranks, and they respect the commutativity between c_ and c+. Consider the eigenvalue problem for the linear operators c_ and c+. The corresponding characteristic polynomial is (—i)ntn_rk* (tr — i)k*, and this gives us the characteristic equation
in-rfc* fi (t - sa)fc* = 0.
a=1
Therefore, the spectra of the eigenvalue problems corresponding to the mappings c_ and c+ consist of the zero eigenvalue of algebraic multiplicity n — rk* and nonzero eigenvalues being powers of the rth root of unity of algebraic multiplicity k*. To see this, it is sufficient to use the transformations (2.6), (2.7) with a special choice of the mappings n_ and n+ and recall the fact that eigenvalues of similar matrices do coincide.
The eigenvalue problem relations
= s-yb), = sfy(6), b = 1,2,..., r, (3.1)
are fulfilled with the eigenvectors represented by n x k* matrices
where its block submatrices are
with constant ka x k* submatrices £a satisfying the conditions
C-aia = £a+1; C+a£a+1 = ^a-
Denote by p the rank of the matrix c_. Then we have n — p = dimker c_. It means that there are an n x (n — p) matrix u and an (n — p) x n matrix vv, corresponding to the zero eigenvalue of c_,
c_u = 0, vv c_ =0. (3.2)
Note that u and vv are orthogonal to for every b = 1,2,..., r. In general, however, the algebraic multiplicity of an eigenvalue does not coincide with its geometric multiplicity, the former is just non less than the latter, and so, n—p ^ n — rk*. In particular, it is exactly what happens to the zero eigenvalue of c_, so that its algebraic multiplicity we have received from the characteristic equation, does not coincide with the dimension of the corresponding null subspace. Hence, one should remember that the rank of c_ might be greater than the total number of its nonzero eigenvalues. It is a consequence of the fact that c_ contains a non-diagonalizable part, corresponding to the zero eigenvalue, that is actually the nilpotent part of c_ according to its Jordan normal form. It is clear that for the case under consideration p = rankc_ = a=1 min (ka, ka+i) ^ rk*.
Let v be the vector dual to the left null vector vv of the matrix c_, and uv the dual of its
right null vector u. They are n x (n — p) and (n — p) x n matrices subject to the conditions
vvv
— In—p and uvu = In_p. Treating c_ as a matrix of a linear operator acting on an n-dimensional vector space V, that is c_ : V ^ V, we see that the latter can be decomposed into a direct sum as V = V0 © V1, where V1 is an rk*-dimensional subspace spanned by the eigenvectors of c_ with nonzero eigenvalues, actually, V1 = im c_; and V0 is simply defined to be its orthogonal complement. Besides, we can perform another decomposition of V, namely V = U0 © U1, where U0 = ker c_, and so, it is spanned by the columns of the n x (n — p) matrix u, while U1 is just the orthogonal complement to U0. These decompositions induce dual decompositions V* = V0* © V* and V* = U0* © U*, such that V* is spanned by the left eigenvectors of c_ being dual to and V0* is determined to be the orthogonal complement to Vf; further, U* is spanned by the left null vectors (the rows of vv) of c_, and U* appears to be its orthogonal complement.
For the case under consideration the linear space V is isomorphic to its dual V * .In view of the above discussion of properties of the eigenvalues of c_, we see that n — rk* = dim V0 ^ dim U0 = n — p, and p = dim U1 ^ dim V1 = rk*. It is clear also that the following relations hold: U0 C V0, V1 C U1 (and similarly for the duals). We stress on that U0 and uq are the right and left null subspaces of the linear operator c_, while V1 and V* are subspaces generated by its right and left ^-eigenspaces. Actually, one can write
vq n uq = uq, V0* n U0* = U0*, V1 n uq = V1* n U0* = 0, (3.3)
Vi n Ui = Vi, V* n U* = V*, Vo n Ui = 0, V0* n U* = 0. (3.4)
These relations are the basis for the subsequent construction of certain useful projectors. A similar treatment can also be given to the linear operator c+ as well.
4. General current conservation laws
Let us introduce a decomposition of the identity operator idV,
In = ni + no,
where n is a projector onto the subspace V, while no is the projector onto the complementary subspace Vo,
n0 =no, n =ni, nino = no ni =0.
We also have
[c-, ni]=0, [c-, no] = 0.
Note that the generalized non-Abelian analogue of the cyclicity property reads here simply c- = c+ = n1. Further, we introduce projectors onto the null subspace Uo and its dual space
U * Uo,
Ro = uuv, Lo = vvv,
such that, together with their orthogonal complements Ri = In — Ro and Li = In — Lo, they reveal the following properties:
Lini =ni, Loni =0, Lono = Lo, (4.1)
niRi =ni, niRo =0, noRo = Ro, (4.2)
and also
Lino = Li — ni =no — Lo, no Ri = Ri — ni = no — Ro. (4.3)
These relations are a direct consequence of (3.3) and (3.4). The r.h.s. of (4.3) would be identically zero, if the nilpotent part of the matrix c_ is trivial. However, in general, these are some nontrivial projectors onto Vo* n U* and Vo n Ui, respectively. The matrix Toda equation (2.3) decomposes into four sets,
(ni7-id_7ni) = [c_, niy-ic+7ni], (4.4)
and besides,
(noY-i d-yn^ = [c-, noy-i c+Yno], (4.5)
(niY-id-Yno) = [c-, niy-ic+Yno], (4.6)
(noY-id-Yni) = [c-, noY-ic+Yni]. (4.7)
It is well known that the zero-curvature condition (2.1) can be interpreted as the integrability condition imposed on the so-called linear problem [3]
t-(A)tf = 0, t+(A)tf = 0,
for some function ^ taking values in G, with the differential operators t- and t+ being explicitly
t-(A) = d- + w-(A), t+(A) = d+ + w+(A).
Indeed, from the requirement
[MA),MA)]tf = 0,
which holds at any power of the parameter A, we derive the Toda equation (2.3).
Consider the parts (4.4)-(4.7) of the Toda equation in their order. The equation (4.4) follows from the integrability condition for the linear problem
(d_ +niY-1 d-Yn + A-1c_) = (d+ + Ani7^c+Y^) ^ = 0. (4.8)
It is actually this, and only this, part of the full matrix Toda equation that is connected with the diagonalizable part of the matrix c_. We are interested in conservation laws respecting the Toda equation (4.4), and thus the linear problem (4.8). Choose k * x ka matrices Ca to be dual to the vectors £a, to have
Ca Ca = Ik*
for every value of a = 1,2,..., r. Introduce n x n matrices D and Dv being explicitly of the forms
D = r_1/2 (V1)ev,^(2)&V.,^(r)CrV) , (4.9)
Dv = r_1/2 (^(-1)CV, ^(-2)CV,..., ^(-r)CrV), (4.10)
and possessing remarkable properties
DVD = DDV = n1,
so that one has
DVn1 = DV, n1D = D.
The matrices D and DV have inherited the block matrix structure induced by the grading condition imposed to obtain the Toda system under consideration. Explicit expressions of the corresponding ka x kb submatrices are
Dab = r_1/2SabCaC6V, DaV6 = r_1/2 S^CaCV
Then the linear problem (4.8) is equivalent to the relations
(d_ + Dvy_1 d_YD + A_1 DVc_D) $ = 0, (d+ + ADvy_1c+yD) $ = 0, (4.11)
where $ = Dv^. Point out that ^(b) = Sb^(0), where S is the diagonal matrix (2.9). Recalling the relations (3.1) and using the expressions (4.9), (4.10) and the property (2.8) of c_, such that
c_D = DS_1,
one can write
c_ ^ $_ = Dvc_D = nS_1.
We conclude that the action of the matrices D and Dv diagonalize the matrix c_. Represent the general solution to the linear problem (4.11) as follows (cf. [8]):
$ = $xexp (—A_1c_z_), (4.12)
where x is a block diagonal matrix, and $ allows for the asymptotic expansion
$ = ^ Ak$k, (4.13)
k>0
such that = while only off-diagonal blocks are nonzero in all other . We also assume that $ and x are such matrices that
[$, ni] = 0, [x, ni] = 0.
It follows from (4.11) that
+ (Dv7-1 ô_7_D)$ + $(d_ xx-1) + A-1[ni S-1, $] = 0, (4.14)
+ $(d+xx-1) + A(Dv7-1c+7D)$ = 0. (4.15) Further, using the asymptotic expansions
d-xx-1 = ^ AkSfe, d+ XX-1 = E Akefc, (4.16) k^ü k^ü
we obtain recurrent relations
k
д-Фк + ^ Ф^ + Dv7-1ô-7D$k + [niS-1, Фк+1]=0, (4.17)
= 1,j=0 i+j=k
k
d+Фк + E + DV7-1c+7D$fc-i =0, (4.18)
i=1,j=0 i+j=k
which allow us to define off-diagonal blocks of Фк+1 (which are, in fact, the only nonzero ones) through those of , l ^ k. And besides, we have
Sk = — Diag (dv7-1d-7D$fc) , ©fc = — Diag (dv7-1c+7D$fc-i) ,
where Diag means taking the block submatrices attached to the main diagonal according to the z-gradation. In particular, we get from (4.17) an explicit form of Ф1 in terms of its block ka x kb submatrices,
„a+b
= b^rf-'d-rfD)., а фЪ.
We see from (4.16) that the block diagonal matrices £k and ©k satisfy the equations
k
d+Sfc — d-©fc + ^ S ©j] = 0.
i,j =0 i+j = k
In terms of the block submatrices the latter equation reads
k
ад—d-©a + j] [sa, ©a ] = o, (4.19)
i,j = 0 i+j = k
where we have taken into account that
(Sk Lb = Sk ^ab, (ek )a,5 = efc ^ab.
Therefore, introducing the quantities
= tr , = tr ©^, we derive from (4.19) r infinite sets of current conservation laws
d+CT a—d-0£ = 0,
where the indices k = 0,1,2,..., and a = 1,2,..., r. For k = 0 we obtain relations which are trivially satisfied due to the Toda equations (2.10). The case k = 1 gives us the energy-momentum conservation law for the non-Abelian Toda system under consideration.
There are other r infinite sets of current conservation laws in the system under consideration. They can be obtained along the same way of approach, only that starting with the Toda equation in the form (2.4). There one would face the diagonalization of the matrix c+ in the corresponding linear problem,
c+ ^ c+ = Dvc+D = n S,
and work recurrent relations out of asymptotic expansions in A-i.
Now, it follows from (4.1), (4.2) that nontrivial conservation laws corresponding to the Toda equation (4.4) are exhausted by our consideration above. Note that the semisimple part of the matrix c- entered this equation alone, while its nilpotent part turned out to be involved and separated into the remaining equations.
The equation (4.5) follows from the integrability condition imposed on the linear problem
(d- + noY-id-Yno + A-ic-) p = 0, (d+ + Anoy-1c+Y^) p = 0,
and to this equation correspond W-symmetry and WZNW-type conservation laws present in the Toda system under consideration. To see this, one can simply use the relations (3.2) and (4.1), (4.2) with the projectors. One derives from (4.5), in particular, that
d+(LoY-1d-YRo) = 0.
To the rest of the matrix Toda equation (2.3) Drinfeld-Sokolov techniques are applicable and correspond certain W-symmetry type conservation laws [5]. Consider the equations (4.6) and (4.7). These are obtained from the integrability condition imposed on the linear problems
(d- + niY-1d-Yno + A-1c-) 0 = 0, (d+ + AniY-1c+Yno) 0 = 0 and ( ) ( )
(d- + noy-1d-Yni + A-1 c-) 0 = 0, (d+ + Anoy-1c+Yn^ 0 = 0, respectively. It follows from these conditions also that
d- (niY-1c+Yno) = 0, d- (noY-1c+Yni) = 0.
Other relations can be obtained while using properties of the projectors.
Concluding this part, we have seen that to the equations (4.6) and (4.7) correspond conservation laws with the conserved matrix-valued current n1 Y-1c+Yno and noY-1c+Yn1 having no dependence on z+. These quantities produce conserved charges as just it happens in non-Abelian Toda systems associated with finite-dimensional Lie groups [5]. Such a mixture of essentially different types of conservation laws in non-Abelian Toda systems can be explain
by certain properties of the matrix-valued mapping entering the Toda equation. Indeed, the matrices c_ and c+ turn out to be a sum of commuting nilpotent and semi-simple parts, that is just implying the so-called Jordan decomposition of a linear operator, for which one obtains W-symmetry and WZNW-type and "usual" current conservation laws' type relations, respectively.
An interesting special case is also revealed when only submatrices C_r and C+r are nontrivial, thus leading to a combination of WZNW- and W-type symmetries.
5. Abelian Toda system
It is instructive to consider an Abelian affine Toda system being a particular case of the non-Abelian system considered in the preceding sections. To construct one, we put r = n and all ka = 1, reproducing the standard gradation of l(gln(c)). The mapping 7 here is a diagonal n x n matrix 7 = Hr5ij||, where r are ordinary functions of z_ and z+. The mappings c_ and c+ can be chosen being proportional to the cyclic elements of g. Explicitly, it reads c_ = H^.j+^l and c+ = H^+^j||, where 5ij is the n-periodic Kronecker symbol and ^ some nonzero constant. We see that the matrix Toda equation (2.3) is equivalent to a system of nonlinear partial differential equations, which is convenient to treat as the infinite system
d+(r_1 d_rj = Vcr"1^ - r^), i e z,
with r subject to the periodicity condition ri+n = T^.
Remembering the basic property (2.8) of c_, where now Sj = si5ij, s = e2ni/n, and introducing n-dimensional vectors ^(1) = S1 ^(0), for the n-dimensional vector ^(0) given by = n_1/2 (1,1,..., 1), we obtain
c_ = ^s_1S1 ^(0) = ^s_V(1).
Representing the n x n matrix D as D = (^(1),^(2),... ,^(n_1),^(0)), that, in terms of the matrix elements, reads Dj = n_1/2sij, we derive the relation
c_D = ^DS _1.
The matrix D is non-degenerate in this Abelian case. Hence, from the relation
c_ — 7_ = D_1c_D = ^S_1
we conclude that the operator D diagonalizes the cyclic matrix c_. Now we have the linear problem with transformed operators ¿1(A) = d_ + 7_1 d_7 + A_1c_, where 7 = 7D, and so, 7ij = n_1/2r^sij, 7_1ij = n_1/2s_ijr"1, and 72(A) = d+ + A7_1c+7. Further, we have 7 = D_1^.
Following again [8], we separate the diagonal and non-diagonal parts of 7 as follows. Represent 7 in the form 7 = $xexp(-A_17_z_), where x is a diagonal n x n matrix, while $ allows for the asymptotic expansion in A (4.13), where again $0 is the unit matrix In, and in all other $k, k ^ 1, only off-diagonal matrix elements are nonzero. Note that 7_ commutes with x. It is convenient to represent x as x = exp a, where a can be asymptotically expanded a = k^0 Akak. Substituting the asymptotic expansions of $ and a to the integrability condition (4.14), (4.15) and taking into account that in the Abelian case n1 — In and Dv — D_1, we obtain the recursive relations
k
d_$k + ^ $id_aj + 7_1d_7$k + [7_,$k+1] = 0, (5.1)
i=1,j=0 i+j = k
k
+ £ $id+<Tj + 7_1c+7$k_i =0 (5.2)
i=1,j=0 i+j = k
for the non-diagonal elements, and
d_ak = —diag (7_1d_7$^ , d+ak = -diag (;y_1c+7$k_0 • (5-3)
For the matrix elements of ak holds the representation (ak)j = a^S^-. These recurrent relations can be resolved for all $k and ak, k = 0,1,2,..., for which it is sufficient to note that $0 = In and
cLc0 = —diag (7_1d_7) , d+a0 = 0. The latter is compatible because
d+ (7_1d_7)ii = 0
thanks to the Toda equations. Taking k = 1 as the first nontrivial example, we find
= — J dz_(7_1d_7$1— J dz+(7_1 c+7,
with explicit forms of the matrix elements under integration
1 n j n
(7-^-7), = ^E^'rr1«-^ (7-V7), = ^E^rr1^!. 1=1 1=1
and where
si+j n
Thus resolving the recurrence (5.1), we can determine $k and so obtain ak from (5.3). For the latter quantities, imposing d+d_ak = d_d+ak, we get current conservation laws
d+S+ — d_S__ =0, (5.4)
where
S+ = diag (7_1d_7$k) , S_ = diag (7_1c+7$k_0 . (5.5)
But then, the relations (5.4) give n infinite sets of current conservation laws, with the currents components (5.5). So for k = 0 these relations are true due to the Toda equations, while for k = 1, seeing the explicit forms of a1 and given above, we find the energy-momentum conservation law for the Abelian affine Toda system.
There are other n infinite sets of current conservation laws corresponding to the Toda equations written in the right invariant form, which is obtained from the linear problem where the matrix c+ is being diagonalized, c+ ^ 7+ = D_1c+D = ^S. Acknowledgements. We are grateful to A. A. Makhnev for discussions.
References
1. Arnold V. I. Mathematical methods of classical mechanics.—Berlin: Springer-Verlag, 1989.—536 p.— (Graduate Texts in Mathematics).
2. Faddeev L. D., Takhtajan L. A. Hamiltonian methods in the theory of solitons.—Berlin: Springer-Verlag, 1987.—592 p.—(Springer Series in Soviet Math.).
3. Zakharov V. E., Shabat A. B. Integration of nonlinear equations of mathematical physics by the inverse scattering method. II // Funkt. Anal. & Pril—1979 —Vol. 13.—P. 13-22.—(in Russian).
4. Leznov A. N., Saveliev M. V. Group-theoretical methods for the integration of nonlinear dynamical systems // Progress in Physics.—Berlin: Birkhauser Verlag, 1992.—Vol. 15.
5. Nirov Kh. S., Razumov A. V. W-algebras for non-Abelian toda systems // J. of Geometry and Physics.— 2003.—Vol. 48.—P. 505-545.—[arXiv:hep-th/0210267].
6. Mikhailov A. V., Shabat A. B., Yamilov R. I. Symmetry approach to classification of nonlinear equations. Complete lists of integrable systems // Russ. Math. Surv.—1987.—Vol. 42.— P. 1-63.
7. Mikhailov A. V., Shabat A. B., Sokolov V. V. The symmetry approach to classification of integrable equations // What is Integrability?— Berlin: Springer-Verlag, 1991.—P. 115-184.
8. Mikhailov A. V. The reduction problem and the inverse scattering method // Physica.—1981.—Vol. 3D.— P. 73-117.
9. Nirov Kh. S., Razumov A. V. On Z-gradations of twisted loop Lie algebras of complex simple Lie algebras // Commun. Math. Phys.—2006.—Vol. 267.—P. 587-610.—[arXiv:math-ph/0504038].
10. Semenov-Tian-Shansky M. A. Integrable systems and factorization problems // Factorization and Integrable Systems.—Boston: Birkhauser, 2003.—P. 155-218.—[arXiv:nlin.SI/0209057].
11. Razumov A. V., Saveliev M. V. Lie algebras, geometry and Toda-type systems.—Cambridge: Cambridge University Press, 1997.
12. Razumov A. V., Saveliev M. V. Maximally non-Abelian Toda systems // Nucl. Phys.—1997.— Vol. B494.—P. 657-686.—[arXiv:hep-th/9612081].
13. Razumov A. V., Saveliev M. V. Multi-dimensional Toda-type systems // Theor. Math. Phys.—1997.— Vol. 112.—P. 999-1022.—[arXiv:hep-th/9609031].
14. Nirov Kh. S., Razumov A. V. On classification of non-Abelian Toda systems // Geometrical and Topological Ideas in Modern Physics.—Protvino: IHEP, 2002.—P. 213-221.—[arXiv:nlin.si/0305023].
15. Nirov Kh. S., Razumov A. V. Toda equations associated with loop groups of complex classical Lie groups // Nucl. Phys.—2007.—Vol. B782.—P. 241-275.—[arXiv:math-ph/0612054].
16. Nirov Kh. S., Razumov A. V. Z-graded loop Lie algebras, loop groups, and Toda equations // Theor. Math. Phys.—2008.—Vol. 154, № 3.—P. 385-404.—[arXiv:0705.2681].
Received September 29, 2010.
Nirova Marina Sefovna Kabardino-Balkarian State University Mathematical Department, associate professor RUSSIA, 360004, KBR, Nalchik, Chernyshevsky St. 175 E-mail: [email protected]
О ЗАКОНАХ СОХРАНЕНИЯ В АФФИННЫХ ТОДОВСКИХ СИСТЕМАХ
Нирова М. С.
С помощью некоторого матричного разложения и проекторов мы показываем, что неабелевы то-довские системы, связанные с группами петель, обладают бесконечными наборами сохраняющихся величин, порождаемых существенно различными законами сохранения токов.
Ключевые слова: неабелевы уравнения Тоды, группы петель, симметрии и законы сохранения.
