The Soliton Equations associated with the Affine Kac-Moody Lie Algebra G_2^{(1)}

0.17 MB·English

by  Paolo Casati

The Soliton Equations associated with the
(1)
Affine Kac–Moody Lie Algebra G .
2
7
0
0
2 Paolo Casati♯, Alberto Della Vedova♭, Giovanni Ortenzi⊙

n
a
J
9 ♯⊙ Dipartimento di Matematica e Applicazioni

Universit`a di Milano-Bicocca
]
I
S Via R. Cozzi, 53
.
n 20125 Milano, Italy
i
l
n ⊙ Dipartimento di Matematica
[
Politecnico di Torino

2
v Corso Duca degli Abruzzi, 24
4
10129 Torino, Italy
3
0
♭ Dipartimento di Matematica
0
1 Universit`a di Parma
6
0 Viale G. P. Usberti, 53/A
/
n 43100 Parma, Italy
i
l
n
E–mail: ♯ [email protected]
:
v
♭ [email protected]
i
X
⊙ [email protected]
r
a
Abstract
We construct in an explict way the soliton equation corresponding to the
affine Kac–Moody Lie algebra G(1) together with their bihamiltonian struc-
2
ture. Moreover theRiccati equationsatisfiedbythegeneratingfunctionofthe
commuting Hamiltonians densities is also deduced. Finally we describe a way
todeducethebihamiltonianequationsdirectlyintermsofthislatter functions
1 Introduction
One of the most fascinating discoveries of the last decades is surely the deep and
fundamental link between the affine Kac–Moody Lie algebras (and their groups as
well), and the soliton equations. This relation first studied and described under
different points of view in a sequel of seminal papers by Sato [18, 19] Date, Jimbo,
Kashiwara and Miwa [11], Hirota [13] Drinfeld and Sokolov [12] and Kac and Waki-
moto [16] has inspired almost innumerable further investigations and generalizations
(see for example the quite interesting papers of Burroughs, de Groot, Hollowood,
Miramontes [1] [2]). Nevertheless, as far as we know, it seems that no explicit de-
scription ofthe hierarchy corresponding in thescheme of DrinfeldandSokolov to the
(1)
affine Kac–Moody Lie algebra G (even of the first non trivial equations) can be
2
found in the literature, fact probably related to the size of standard realization of G
2
(namely by 7×7 matrices). The aim of this letter is to fill this gap and to show how
the bihamiltonian formulation of the Drinfeld-Sokolov reduction [8] [9] [4] makes the
computations involved more reasonable. The main ingredient of our construction
will be indeed the technique of the transversal submanifold, which can be imple-
mented only in the bihamiltionian reduction theory, and which drops drastically the
free variables involved in the computations. The same technique provides also a way
to construct a Riccati type equation for the formal Laurent series for the conserved
quantities of the corresponding integrable system. Since this equation at least in
principle may be iteratively solved in a pure algebraic way, the bihamiltonian tech-
nique offers a computational way to construct the whole hierarchy. Moreover what
happens in the case of the affine Lie algebras A(1) suggests that it could be exists
n
a way to obtain directly the equations of the hierarchy,starting by such conserved
quantities, without referring directly to the underlying bihamiltonian structure.
The paper is organized as follows in the first section we perform the bihamilto-
nian reduction of the Drinfeld-Sokolov hierarchy defined on the affine Kac–Moody
(1)
Lie algebra G obtaining the reduced bihamiltonian structures and the first equa-
2
tions of the hierarchy as well. In the second and last section we explain and perform
the so called Frobenius technique for the same algebra obtaining a so called Riccati
equation satisfied by the generating function of the conserved densities. Finally we
shall show how the knowledge of this function is enough to construct the entire
hierarchy of bihamiltonian equation.
Aknowledgements We would like to thank Marco Pedroni and Youjin Zhang for
many useful discussions on the subject.
1
2 The Bihamiltonian Reduction Theory of the Lie
(1)
algebra G
2
The aim of this first section is to obtain the bihamiltonian structure of the soliton
(1)
equation associated with the Kac–Moody affine Lie algebra G in the Drinfeld–
2
Sokolov theory by performing a bihamiltonian reduction process.
For the convenience of the reader, let us start by recalling the main facts of the
bihamiltonian reduction theory, referring for more details to the papers [8] [9] [4]
where this theory was first developed. A bihamiltonian manifold M is a manifold
equipped with two compatible Poisson structures, i.e., two Poisson tensors P and
0
P such that the pencil P = P − λP is a Poisson tensor for any λ ∈ C. Let us
1 λ 1 0
fix a symplectic submanifold S of P and consider the distribution D = P Ker(P )
0 1 0
then the bihamiltonian structure of M, provided that the quotient space N = S/E
is a manifold, can be reduced on N ([8] Prop 1.1).
To construct the reduced Poisson pencil PN from the Poisson pencil P on M
λ λ
we have to perform the following steps [6]:
1. For any 1–form α on N we consider the 1–form α∗ on S, which obviously
belongs to the annihilator E0 of E in T∗S.
2. We construct a 1–form β on M which belongs to the annihilator D0 of D and
satisfies
i∗β = π∗α (2.1)
S
(i.e., a lifting of α).
3. We compute the vector field P β, which turns out (see [6] Lemma 2.2) to be
λ
tangent to S.
4. We project P β on N:
λ
(PN) α = π (P ) β.
λ π(s) ∗ λ s
Weshallnotcomputeinthenextsectionthereducedbihamiltonianstructurerelated
(1)
to the affine Kac–Moody Lie algebra G using directly the above cited Theorem
2
but rather implementing the technique of the transversal submanifold [9] and [6] in
order to avoid most of the computations involved.
A transversal submanifold to the distribution E is a submanifold Q of S, which
intersects every integral leaves of the distribution E in one and only one point. This
condition implies the following relations on the tangent space:
T S = T S ⊕E ∀q ∈ Q (2.2)
q q q
The importance of the knowledge of a transversal submanifold lies in the following
Theorem proved in[8] [4]:
2
Theorem 2.1 Let Q be a transversal submanifold of S with respect the distribution
E. Then Q is a bihamiltonian manifold isomorphic to the bihamiltonian manifold
N and the corresponding reduced Poisson pair is given by:
PQ α = Π (P ) α˜) i = 0,1 (2.3)
i ∗ i q
(cid:16) (cid:17)q
where q ∈ Q, α ∈ T∗Q Π : T S → T Q is the projection with respect the decompo-
q ∗ q q
sition (2.2) and α˜ ∈ T∗M satisfies the conditions:
q
α˜ = 0 α˜ = α. (2.4)
|Dq |TqQ
Actually for our porpoises the hypothesis ofthis Theorem may be slightly relaxed by
considering a submanifold Q which is only locally transversal (i.e., it satisfies only
the weaker condition (2.2)) in this case of course Q and N could be only locally
isomorphic (see [17] for more details).
The bihamiltionian manifolds which are interesting in, are the bihamiltonian
manifold naturally defined on the affine Kac–Moody Lie algebras. An affine non
twistedLiealgebragcanberealizedasasemidirect productofthecentralextensions
of a loop algebra of a simple finite dimensional Lie algebra g and a derivation d:
b
g = C∞(S1,g)⊕Cd⊕Cc.
Then the Lie bracket of twob(typical) elements in g of the type
X = x ⊗xn +µ c+ν d, Y = xb ⊗xm +µ c+ν d
n 1 1 m 2 2
with x ,y ∈ g and n,m ∈ Z, µ ,µ ,ν ,ν ∈ C is
n n 1 2 1 2
[X,Y] = [x ,y ]⊗xn+m +(x ,y )cδ −nν x ⊗xn +mν y ⊗xm (2.5)
n m n m n+m,0 2 n 1 m
where [x ,y ] is the Lie bracket in g, and (·,·) is the killing form of g. (and finally
n m
δ is the usual Kronecker delta). In what follows the derivation d will not play any
role. Being g a affine (infinite dimensional) manifold we may identify it with its
tangent space at any point. Moreover using the non degenerated form
b
1
h(V ,a),(V ,b)i+ (V (x),V (x))dx+ab (2.6)
1 2 1 2
Z
S
we may identify at any point S the tangent space with the corresponding cotangent
space T M = T∗M. Using these identifications we can write the canonical Lie
S S
Poisson tensor of g as
P (V) = c∂ V +[S,V]. (2.7)
b (S,c) x
It can be easily shown that this Poisson tensor is compatible with constant Poisson
tensor obtained by freezing the tensor (2.7) in any point of M. In particular the
hierarchies of Drinfeld and Sokolov turns out to be bihamiltonian with respect to
the bihamiltonian pair P , P where P is the canonical Poisson tensor (2.7) and P
1 0 1 0
is the constant Poisson tensor
(P ) (V) = [A,V]. (2.8)
0 (S,c)
where A is the constant function of C∞(S1,g) whose value is the element of minimal
weight in g.
3
3 The reduction process
In this section, following [9], we perform the bihamiltonian reduction of the excep-
(1)
tional Lie algebra G . It is a rank 2 simple Lie algebra whose Cartan matrix is
2
2 −3
 
. A possible Weyl basis is:
 −1 2 
 
 
H = d −d +d −d H = d −d +2d −2d +d −d
1 22 33 55 66 2 11 22 33 55 66 77
E = d +d E = d +d +2d +d (3.1)
1 23 56 2 12 34 45 67
F = d +d F = d +2d +d +d
1 32 65 2 21 43 54 76
where d is a 7×7 matrix with 1 in the ij position and zero otherwise.
ij
Thus the elements of the algebra has the form
h e −e 2e −6e 6e 0
2 2 3 4 5 6
 
 
 f h −h e e −2e 0 6e 
 2 1 2 1 3 4 6 
 
 
 
 f f −h +2h e 0 −2e 6e 
 3 1 1 2 2 4 5 
 
 
 
v =  4f −2f 2f 0 2e −2e 4e .
 4 3 2 2 3 4 
 
 
 
 6f −2f 0 f h −2h e e 
 5 4 2 1 2 1 3 
 
 
 
 6f 0 −2f f f −h +h e 
 6 4 3 1 1 2 2 
 
 
 
 0 6f −6f 2f −f f −h 
 6 5 4 3 2 2 
 
(1)
As already noted on G is defined a bihamiltonian structure given by canonical
2
Lie Poisson tensor and by the tensor (2.8) where in the present case the element of
minimal weight is A = F .
6
To perform the Marsden-Ratiu reduction process of such bihamiltonian structure
we need to compute first a symplectic leaf S of P and second the distribution E =
0
P (Ker(P ) on the point of S. As proved in [9] the symplectic leaves of the constant
1 0
(1)
Poisson tensor are affine subspaces modelled on the subspace of G orthogonal to
2
the isotropic algebra of the element A. Following Drinfeld and Sokolov let us choose
that passing through the point b = E +E :
1 2
S = b+h(t)(2H +H )+f (t)F +f (t)F +f (t)F +f (t)F +f (t)F .
1 2 1 1 3 3 4 4 5 5 6 6
4
Then the (constant) distribution E = P kerP evaluated on the points of S is
1 0
t 0 0 0 0 0 0
1
 
 
 0 t 0 0 0 0 0 
 1 
 
 
 
 t t 0 0 0 0 0 
 2 3 
 
 
 
 2t −2t 0 0 0 0 0 .
 4 2 
 
 
 
 t −t 0 0 0 0 0 
 5 4 
 
 
 
 t 0 −t t t −t 0 
 6 4 2 3 1 
 
 
 
 0 t −t t −t 0 −t 
 6 5 4 2 1 
 
Luckily enough we may apply Theorem 2.1 since the submanifold Q of S
0 1 0 0 0 0 0
 
 
 0 0 1 0 0 0 0 
 
 
 
 
 0 u 0 1 0 0 0 
 0 
 
 
 
Q =  0 0 0 0 2 0 0 
 
 
 
 
 0 0 0 0 0 1 0 
 
 
 
 
 6u 0 0 0 u 0 1 
 1 0 
 
 
 
 0 6u 0 0 0 0 0 
 1 
 
is transversal to E.
The actually computations of the explicit form of the reduced Poisson pencil as
observed in [5] boils down to find (given a 1-form v = (v ,v ) ∈ T∗(Q)) a section
0 1
V(v) in i∗ (T∗M) (where i Q ֒→ M is the canonical inclusion) such that P V(v) ∈
Q Q λ
TQ. This implies that the entries of V(v) are polynomials functions of the elements
(8e ,288e ) and that the reduced Poisson pencil
1 6
dq
= (PQ) v = V(v) +[V(v)+λa,q],
dt λ q x
λ
becomes
5
du 7 3 1 11 5 5 1
0 = β u′v(4) + u v′ − (u′)2v′ + u′′v′′′ + u(4)v′ + u′′′v′′ + u v(5)+
dt (cid:18)32 0 1 4 1 1 8 0 1 32 0 1 32 0 1 16 0 1 16 0 1
λ
5 1 1 1 1 7 1 1
+ u′v − u2v′′′ + u(5)v + u v′ + u′v − v′′′ − v(7) − u u′′v′+
8 1 1 32 0 1 32 0 1 4 0 0 8 0 0 4 0 32 1 8 0 0 1
5 3 1 3
− u u′v′′ − u′u′′v − u u′′′v −λ v′
32 0 0 1 32 0 0 1 32 0 0 1(cid:19) 4 1
du 1 7 1 1 1 3
1 = u′v(8) − (u′′′)2v′ − u4v′′′ + u(9)v − u2v′′′ + u v′+
dt (cid:18)96 0 1 144 0 1 1728 0 1 1728 0 1 32 0 0 4 1 0
λ
1 47 13 5 7
+ u v(5) + u(4)v(5) + u′′v′′′ + u2u′v(4) + u u′u(4)v +
72 1 1 576 0 1 144 1 1 288 0 0 1 864 0 0 0 1
61 1 5 1 1 23
− (u′′)2v′′′ + u v(9) + u(4)v′ + u′v − v(11) − u u(5)v′′+
576 0 1 432 0 1 96 1 1 8 1 0 1728 1 576 0 0 1
1 29 85 1 3 1
+ u v(5) + u′′′v′′ + u(6)v′′′ + u(7)v′′ + u′v(4) − u u′′v′+
16 0 0 288 1 1 1728 0 1 48 0 1 32 0 0 24 1 0 1
1 1 1 23 29
− u u′v′′ − u u v′′′ − u u′v′′ + u′′u2v′′′ + u (u′)2v′′′+
24 0 1 1 36 1 0 1 24 1 0 1 864 0 0 1 864 0 0 1
17 5 1 7 67
− u′′u′′′v′′ − u′′u(4)v′ − u′′u(5)v − u2(u′)2v′ − u′u′′′v′′′+
96 0 0 1 64 0 0 1 72 0 0 1 1728 0 0 1 432 0 0 1
1 1 1 1 1 1
− u(4)u′′′v − u u′v′′ + u2u′v + u2u v′ − u3u′′v′ − u3u′′′v +
48 0 0 1 32 0 0 0 144 0 1 1 72 0 1 1 432 0 0 1 1728 0 0 1
1 1 11 1 5
− u3u′v′′ − u (u′)3v − u u (6)v′ − u u(7)v − u′u(4)v′′+
288 0 0 1 1728 0 0 1 864 0 0 1 576 0 0 1 48 0 0 1
11 5 61 3 35 1
− u′u(5)v′ − u′u(6)v − u u(4)v′′′ + u′′v′′′ + u′′′v(6) + (u′)3v′′+
288 0 0 1 864 0 0 1 864 0 0 1 32 0 0 576 0 1 64 0 1
1 35 1 1 1
− v(7) + u u′u′′′v′ − u u′′v(5) − u′u′v′ + u u′u v +
32 0 864 0 0 0 1 18 0 0 1 36 0 1 1 72 0 0 1 1
1 5 1 1 1
+ u′′′v′′ + u′v(4) − u2v(7) + u u′′u′′′v − u2u′u′′v +
32 0 0 144 1 1 288 0 1 72 0 0 0 1 432 0 0 0 1
11 11 5 5 5
+ u(5)v(4) + u u′u′′v′′ − u u′′′v(4) − u u′′′v − u′u′′v +
144 0 1 144 0 0 0 1 64 0 0 1 288 0 1 1 288 0 1 1
7 1 19 13 17
− u u′′v′ − u′u′′v + (u′)2u′′v′ + u2u′′′v′′ + u2u(4)v′+
144 0 1 1 72 1 0 1 576 0 0 1 576 0 0 1 1728 0 0 1
1 1 13 1 1
+ u′(u′′)2v + u (u′′)2v′ + u′′′(u′)2v + u2u(5)v − u u′′′v +
96 0 0 1 36 0 0 1 1728 0 0 1 576 0 0 1 72 1 0 1
7 5 1 1 1
− u u′v(6) − u′u′′v(4) + u(5)v + u3v(5) + u(8)v′+
288 0 0 1 36 0 0 1 96 1 1 432 0 1 192 0 1
1 7 1 1 1 1
+ u′′v(7) − (u′)2v(5) −λ − u′′′v + u2v′ + v(5) − u′′v′+
32 0 1 192 0 1 (cid:19) (cid:18) 72 0 1 72 0 1 72 1 24 0 1
1 1 1 3
+ u u′v − u′v′′ − u v′′′ + v′
72 0 0 1 24 0 1 36 0 1 4 0(cid:19)
where the prime indicates the space derivative. Having the reduced bihamiltonian
structurewearenowabletowriteexplicitlythefirstnon-trivialflowsofthehierarchy
6
[8]. Since the Casimirs of P are given by the functionals
0
u3
H = dxu and H = dxu′′u − 0 +108u (3.2)
0 ZS1 0 1 ZS1 0 0 3 1
we obtain
1
u = u (3.3)
0,t0 8 0x
1
u = u (3.4)
1,t0 8 1x
and
1
u = − (u(5) +5u′u2 −5u′′′u −5u′′u′ −540u ) (3.5)
0,t1 864 0 0 0 0 0 0 0 1
1
u = − (−9u(5) +15u′′′u +15u′′u′ +10u′u′′ −5u′u2) (3.6)
1,t1 864 1 1 0 1 0 1 0 1 0
4 The Frobenius Technique
In the first section we have found the bihamiltonian structure of the soliton equation
(1)
associated to the affine Kac–Moody Lie algebra G together with its first not
2
trivial equations. This is of course by far not the same thing as to provide a way
to actually compute all the soliton equations of the hierarchy. In the setting of
the bihamiltonian theory this second important problem is tackled by looking for
Casimirs of the Poisson pencil P = P −λP i.e., solutions of the equations
λ 1 0
V +[V,S +λA] = 0 s ∈ S (4.1)
x
which are formal Laurent series V(λ) = ∞ V λ−k whose coefficients are one
k=−1 k
forms defined at least on the points of SPwhich are exact when restricted on S.
Indeed once such a solution is found the vector fields of the hierarchy can be written
in the bihamiltonian form X = P V = P V .
k 0 k 1 k−1
This latter task is unfortunately usually a very tough problem, but in the contest
of the integrable systems defined on affine Lie algebra it can be solved by using the
generalization of the dressing method of Zakharov Shabat proposed by Drinfeld and
Sokolov [12]. Unfortunately exactly as happens for the Drinfeld–Sokolov reduction
(1)
for the Lie algebra G the computations involved to derive the explicit expression
2
of the bihamiltonian fields of the hierarchy are very complicated. The aim of this
last section is to show how the so called Frobenius technique ([7]) provides somehow
a shortcut of the Drinfeld–Sokolov procedure.
More precisely this technique will give a way to compute algebraically by a
recursive procedure the conserved densities of the hierarchy and therefore the corre-
sponding (maybe without passing throughthePoisson tensors) bihamiltonian vector
fields as well. However implementing such technique requires to give up the pure
7
geometrical description of the hierarchy of the first section in order to consider also
the minimal true loop module C∞(S1,R7) of G(1) [10] together with its geometrical
2
dual space and the set of its linear automorphisms as well.
The starting point of the theory is indeed to observe ([5],) that V ∈ T∗S solves
S
(4.1) at the point S ∈ (S,c = 1) if and only if it commutes viewed as linear opera-
tor in End(C∞(S1,C7)) ( up the canonical identification explained in the previous
section) with the linear differential operator −c∂ + S + λA. Although this latter
x
task seems not really easier then the first one, it suggests a way (using the action
of the affine Lie group G on the representation space C∞(S1,R7)) to obtain di-
rectly the equations of the hierarchy together with their hamiltonians. Following
e
what suggested by Drinfeld and Sokolov we can find the elements V commuting
with −c∂ + S + λA using the observation that the element B + λA is a regular
x
element and therefore its isotropic subalgebra g is a Heisenberg subalgebra
B+λA
H of g spanned (up to the central charge) in our representation by the matrices
n n−1
Λ6n+1 = λ (B + λA), Λ6n−1 = λ (B + λA)5 with n ∈ Z. (For sake of
e
24 24 24 24
(cid:16) (cid:17) (cid:16) (cid:17)
simplicity, from now on, we rescale λ → λ). From this fact Drinfeld and Sokolov
24
proves indeed the
Proposition 4.1 For any operator of the form −∂ +S+λA with s ∈ S there exists
x
a element T in G such that:
e
T(−∂ +S +λA)T−1 = ∂ +(B +λA)+H, H ∈ H. (4.2)
x x
Therefore the set of the elements in g commuting with −∂ + S + λA is given by
x
T−1HT.
e
The knowledge of a such a T allows us to compute explicitly for any choice of
an element in H the corresponding hierarchy of vectors fields together with their
Hamiltonians.
Proposition 4.2 Let C = c Λ−j with c ∈ C be an element in H then:
j=±1mod(6) j j
P
1. the element V = T−1CT solves equation (4.1);
C
2. its hamiltonian on S is the function H = hJ,Ci where J is defined by the
C
relation
J = T(S +λA)T−1 +T T−1 (4.3)
x
3. in particular if C has the form C = Λj, j = ±1mod(6) (say j = 6n±1) then
V and H simply denoted respectively Vj has the Laurent expansion
C C
1
Vj = λn V (4.4)
λp+1 6p±1
pX≥−2
Proof .
8
1. It was already proved above.
2. Using equation (4.3) we can rewrite equation (4.2) in the form T(−∂ + S +
x
λA)T−1 = −∂ +J showing the J commutes with C then:
x
d
H = hJ˙,Ci = hTS˙T−1 + T˙T−1,J ,Ci
C
dt h i
but since C commutes with J we get
d
H = hTS˙T−1,Ci = hS˙,T−1CTi = hS˙,V i.
C C
dt
3. Equation (4.4) follows for j = 6n+1 from the identity
res(λp−nVj) = res(λp−nTΛjT−1) = res(λp−nTλnΛT−1)
= res(TλpΛT−1) = res(TΛ6p+1T−1) = res(V6p+1)
while similar computations show that if j = 6n − 1 then res(λp−nVj) =
res(V6p−1).
(cid:3)
As already pointed out the actually computation of the element T (which by the
way provides also the vector fields of the hierarchy) is in our case quite complicated.
To avoid such computations let us first solve the related problem of finding the
eigenvalues of the operator −∂ +S +λA:
x
−ψ +(S +λA)ψ = µψ. (4.5)
x
This latter problem can be solved by the observation that the integral leaves E
are orbits of a group action, completely characterized by the distribution E at the
special point B. It holds indeed:
Proposition 4.3 The subspace g := {V ∈ g |V +[V,B] ∈ g⊥} is a subalgebra
AB A x A
of g contained in the nilpotent subalgebra n of loops with values in the maximal
−
nilpotent subalgebra spanned by the negative (it depends how G is defined). There-
2
fore the corresponding group G = exp(g ) is well defined. The distribution E
AB AB
is spanned by the vector fields (P ) (V) with V belonging to g , and its integral
1 B AB
leaves are the orbits of the gauge action of G on S defined by:
AB
S′ = TST−1 +T T−1. (4.6)
x
Explicitly the for the Lie algebra G the group G is: Now it easily to see that
2 AB
equation (4.6) implies on the space End(C∞(S1,C7) that the linear differential op-
erators −∂ +S +λA and −∂ +S′+λA are conjugated by an element T ∈ G in
x x AB
formula:
(−∂ +S′ +λA)◦T = T ◦(−∂ +S +λA). (4.7)
x x
9