Two Poisson structures invariant with respect to discrete transformation in the case of arbitrary semi-simple algebras

0.23 MB·English

by  A. N. Leznov

Two Poisson structures invariant with respect to
discrete transformation in the case of arbitrary
8
0
semi-simple algebras
0
2

n A. N. Leznov∗
a
J

6
1
Abstract

] Two Poisson structuresinvariant with respect to discrete transforma-
t
a tion of Maximal root presented in explicit form for arbitrary semi simple
l algebra. Thus problem of construction of multi-component hierarchies of
-
p integrable systems is solved.
e
h
[ 1 Introduction

1
v In the present paper we would like to systematize results obtained in the pre-
1 vious papers of the author [1],[2] - [7] concerning the discrete transformations,
4
hierarchies of systems of equations and its multi-soliton solutions of n - wave
5
problem. Now we would like to forget about the origin of the problem and
2
. consider some general property of semi-simple algebra. The problem may be
1
formulated as the following. With each semi-simple algebra it is possible to
0
connect two Poisson structures invariant with respect to some discrete trans-
8
0 formation of the given form. Of course this observation was done only after
: considerationoftheexamplesofthesemisimplealgebrasofthelowranks. And
v
weadvisetothereaderatfirstreadoneofthementionedabovepapersinwhich
i
X calculations may be done by fourth action of arithmetic plus differentiation.
r
a
2 Participants of the game and its rules
2.1 Notations
G - arbitrary semi-simple algebra. R - the space of its positive and negative
roots.Thedimension ofR equal2n=N−r, N - dimensionofthe algebra,r its
rank. XR aregeneratorswith commutationrelations[XR,XR′ ]=DRR,+RR′′XR+R′
where DR+R′ structure constants (some times we omitted sign X → R). f
R,R′ R R
∗UniversidadAutonomadelEstadodeMorelos,CCICAp,Cuernavaca, Mexico
1
- the system of functions depending on one parameter x - space coordinate
±
(and possible some other parameters). Some times we use notation f±R =fR.
Elements of Cartan sub algebra will be denoted by little latine letters c =
c h ,[c,X ] = c X . Generators of all roots of the algebra are normalized
i i R R R
on unity (XR,XR′ =δR,−R′.
P
2.2 The grading of the maximal root
In what follows we use the grading of the maximal root [8]. This means that
all generators of the algebra may be distributed on the subspaces with ±2,±1
and 0grading indexes. The grading index is defined by the proper value of the
Cartan generator of the maximal root H = [X+,X−],[H ,X±] = ±2X±.
M M M M M M
Thus arbitrary element of semi simple algebra may be represented as
f =f(+2+f(+1+f0+f(−1+f(−2, [H ,f(m]=mf(m
M
The subspaces f(±2 = f(±X± are one dimensional. The subspaces with ±1
M M
gradedindexesf(±1 =f(±+f(±maybedecoupledinsuchwaythat[f(±,f(±]=
+ − + −
±
cX and all generators in subspaces with the same additional index ± are
M
mutually commutative. Elements of ofone gradedsubspaces arerepresentedin
aformf(± = fR(+1(S(−1)XR(+1(S(−1). Inwhatfollowszerogradedsubspaceis
limitedbyadditionalcondition[f0,X±]=0. Thezerogradedsubspaceconsists
P M
from the different elements [XR(+1,XS(−1] not coincide with the elements of
Cartan sub algebra. The following Furie decomposition take place
(A(+1X−R(+1(S(−1))(B(−1XR(+1(−S(−1))=(A(+1B(−1)
X
andsomemodifiedformulawithrespecttosummationonelementsofzeroorder
subspace
(A0X−r0)(B0Xr0)=(A0(B0− hβkβ−,1γ(hγB0))=
X X
(A0B0)− (h B0)k−1(h A0)
β β,γ γ
The last modification is connecteXd with the fact that Cartan elements of the
simple roots of the algebra may be among A0,B0 and they must be excluded
from the result of summation on zero order subspace.
2.3 Discrete transformation, Frechet derivative and Pois-
son structure
We repeat corresponding text from [9]. The discrete invertible substitution
(mapping) defined as
u˜=φ(u,u′,...,ur)≡φ(u) (1)
2
uissdimensionalvectorfunction;ur itsderivativesofcorrespondingorderwith
respect to ”space” coordinates.
The propertyofinvertibility meansthat(1)canbe resolvedand”old”func-
tion u may expressed in terms of new one u˜ and its derivatives.
Frechet derivative φ′(u) of (1) is s×s matrix operator defined as
φ′(u)=φu+φu′D+φu′′D2+... (2)
where Dm is operator of m-times differentiation with respect to space coordi-
nates.
Let us consider equation
F (φ(u))=φ′(u)F (u) (3)
n n
whereF (u)iss-componentunknownvectorfunction,eachcomponentofwhich
n
depend on u and its derivatives not more than n order. If equation (3) posses
at least one not trivial solution (Ftrivial ≡ u′) then substitution (1) is called
integrable one.
With each discrete substitution it is possible try to connect Poisson struc-
ture defined as anti symmetric matrix valued operator JT(u) = −J(u) and its
invariance with the respect to discrete transformation (1) is fixed by equation
it satisfied
φ′(u)J(u)(φ′(u))T =J(φ(u)), φ′(u)H(u)(φ′(u))−1 =H(φ(u)) (4)
where(φ′(u))T =φTu−DφTu′+D2(φu′′)T+...andJ(u),H(u)areunknowns×s
matrix operators, the matrix elements of which are polynomial of some finite
orderwithrespecttooperatorofdifferentiation(ofitspositiveandnegativede-
grees). Firstequationin(4) is conditionofinvarianceofPoissonstructure with
respect to discrete transformation, the second one is equation for raising oper-
ator. Two different Poissonstructures lead obviously to H(u)=J(u) J−1(u) .
1 2
3 The statement of the problem
In[1]itwasshownthatequationsofn-thwavesareinvariantwithrespectsome
numberofdiscretetransformations(coincide withr -the rankofcorresponding
semi-simple algebra). Namely with suchkind ofdiscretetransformationwe will
have deal in what follows below. It will be shown that there exist two different
Poisson structures (which can be used for construction of the hierarchies of
integrable systems of equations). But the fact of existence of such Poisson
structures is the inner property of semi-simple algebra and the written ad hoc
discretemappingofthedefiniteform. Authordon’tknowandhavenoguesshow
these two objects may be connected in the frame work of group representation
theory.
3
3.1 Discrete transformation of the maximal root
The discrete transformation of the maximal root [8],[1] is the mapping realized
on the space of f functions depending on r arbitrary parameters [c,X ] =
R R
c X and one space coordinate x (∂f ≡f′)
R R ∂x
TfM+→= 1 , TfM(+→1= [XM+,f(−1] ≡ α(+1, TMf→=(f0+ 1 [α(+1,f(−1]) ,
M f− f− f− 0 2f− h
M M M M
where (..) means factor on Cartan sub-algebra. In other words
h
([α(+1,f(−1]]) ≡[α(+1,f(−1]− h k−1(h [α(+1,f(−1])
h β β,γ γ
X
where h Cartan element of simple root of the algebra. Below we use the
γ
notation θ ≡([X+,f(−1][c,f(−1]))
M
TM→ (f−)′− 1θ 1
[c,f(−1]=(f(−1)′+ M 2 [c,f(−1]+ [c[[α(+1,f(−1]f(−1]]−
− −
c f 6f
M M M
[f(−1,[c,f0]]+f−[[c,f(+1]X−]
M M
c TfM−→=c (f−)2f+ + (fM−)′′ + 1 ([X+,f(−1][c,(f(−1)′])− ((fM−)′− 12θ)2
M M M M M c c M c f−
M M M M
1 1
+ ([α(+1,f(−1][α(+1,f(−1])+f−(f(−1[c,f(+1])− ([α(+1,f(−1][c,f0])
24f− M 2
M
(5)
Wewouldliketoshowthattransformation(5)iscanonicalone. Thatmeans
thatthereexistsomefunctionalG(generatingfunction)dependingonthepairs
TM→
of transformed and initial variables fR ,fR′ in which encoded all informa-
tion about (5). We will use the following identification for general impulses
PM = c f+,P1 = [c,f(+1],P0 = [c,f0] and coordinates X = f−,X =
M M + M M 1
f(−1,X = f0 where f0 means part of f0 connected with positive and neg-
0 − ±
ative spaces of zero order subspace. We choose the following dependence of
TM→ TM→
mentioned above functional G = G(X ,X ;X , P ) then in the case of
M,1 M,1 0 0
canonical transformation
PM,1 =− δG , PTMM→,1= δG , P0 =− δG , TXM→=− δF (6)
δXM,1 δ XTM→ δX0 0 δ TPM0→
M,1
where functional derivatives coincide with the Frechet one (2). In the case of
canonical transformation of the classical dynamics (and thermodynamic) func-
tionalGdependonlyoncanonicalcoordinatesandgeneralizedimpulsesbutnot
on its derivatives. It is not surprising that discrete transformationis connected
4
with generalized canonical ones. This fact was observed some times ago and
proof of this fact with respect to self-dual system ( and many other integrable
systems) reader can find in [10].
We present below explicit expression for density of functional G. Reader
can check by using only one operation of differentiation that (6) and (5) are
identical
(TfM−→) 1 TM→ 1 f′ 1
G=c M )− (α(+1[c,f(−1])− ( M)2− (α(+1[c,(f(−1)′])−
M f− f− 2c f c f−
M M M M M M
1 1
− (α(+1[c,f(−1])2+ ([α(+1,f(−1][α(+1,f(−1])−
− −
8c (f )2 48(f )2
M M M
(TPM0→X )+ 1 ([c[α(+1,f(−1]] X )− (7)
0 − + 0
2f
M
− 1−([α(+1,f(−1]− TPM0→)+ 1−([c[α(+1,f(−1]]+[α(+1,f(−1]−)
2f 4f
M M
As reader can see from (7) generating function in the case under consideration
dependonthederivativesofthefirstorderoff(−1functionsandsecondorderon
R
−
f one. Itleadtosomemodificationinusualtheoryofcanonicaltransformation
M
which will be noticed below.
4 Conservation of the Poisson brackets
ThemainpropertyofcanonicaltransformationconsistsinconservationofPois-
son brackets between dynamical variables involved. This fact proofs automati-
cally in the usual theory (without derivatives of dynamical variables). We will
trytorepeatcorrespondingcalculationandshowthatthispropertywithrespect
to discrete transformation of the previous section is conserved.
For this goal let us consider Jacobi matrix
J = D(TMP→,TXM0→;TMP0→,TMX→) = D(TMP→,TXM0→;TPM0→,TMX→) 10 00 01 00
D(p,x ;p ,x) D(p,p ;x ,x) 0 1 0 0
0 0 0 0
0 0 0 1
 
 (8)
In (8) in P we unite generalized impulses P=P ,P1 and the same do with
M
corresponding coordinates X=X ,X1. We can not do the same with coordi-
M
nates and impulses with zero indexes by the same reason why unite canonical
transformation in usual theory my be written not in all pairs of canonical vari-
ables. For further transformation of (8) we unite ”4” independent variables of
5
TM→ TM→
G functional into two pairs y¯= ( P , X ), y = (x,x ). In this notations (6)
0 0
can be rewritten as follows
TMP→ 0 1 p
TM→ = −1 0 Gy¯ ≡σ2Gy¯, p =−Gy
X0 ! (cid:18) (cid:19) (cid:18) 0(cid:19)
After differentiation of the last equality with respect to pairs of arguments p=
(p,p ),y =(x,x )andtakingintoaccountindependentargumentsofgenerating
0 0
function G we obtain
y¯ =−(G )−1,y¯ =−(G )−1G ,
p y,y¯ x y,y¯ y,y
TM→
TMP→ =−σ2Gy¯,y¯(Gy,y¯)−1,
X !
0 p
TM→
TMP→ =−σ2Gy¯,yσ1+σ2Gy¯,y¯(Gy,y¯)−1Gy,y
X !
0 y
0 1 0 1 1 0
(σ = , σ = , σ = ).
1 1 0 2 −1 0 3 0 −1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
In the general case, when generating functional has arbitrary dependence on
derivatives of p,x arguments the last expression may be had no mean, but in
the case under consideration generating functional is linear in y¯arguments. In
this case matrix G depend only on generalized coordinates X and impulses
y,y¯
P but not from their derivatives and all operations done above are absolutely
correct. After this manipulations expression for Jacobi matrix valued operator
may rewritten in two dimensional form ( we take into account condition of
linearity of G,G =0
y¯,y¯
1 0 0 0
0 σ G σ 0 0 1 0
J = 3 y¯,y 1 (=δ(2→3))
−(G )−1 −(G )−1G σ 0 1 0 0
y,y¯ y,y¯ y,y 1
(cid:18) (cid:19)
0 0 0 1
 
 
0 σ 0 σ
Now by direct calculations let us check that J 3 JT = 2 .
σ 0 σ 0
3 2
(cid:18) (cid:19) (cid:18) (cid:19)
Indeed
0 σ 0 I
σ δ(2→3) 3 δ(2→3)TσT =
1 −σ 0 1 −I 0
3
(cid:18) (cid:19) (cid:18) (cid:19)
and
0 I 0 σ G G σ GT (G )−1
J JT = 3 y¯,y y,y¯ 3 y,y y¯,y =
−I 0 −(G )−1 −(G )−1G 0 −(G )−1
y,y¯ y,y¯ y,y y¯,y
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
6
0 σ
3
−σ G )−1(GT −G )(G )−1
(cid:18) 3 y,y¯ y,y y,y y¯,y (cid:19)
In the case when G don’t depend on space derivatives the term in the right
dawncornerofthelastmatrixobviouslyequaltozero. Inthecaseoffunctional
(7) the checking of this fact reader find in Appendix.
Inthepreviouspapersitwasusedtheorderoffunctionsf+,f(+1,f0,f0f(−1,f+
M + , M
Tocometosuchorderfromconsideredaboveitisnecessarytodosomenumber
of point like tangent transformation: change by the places X → P and do
0 0
mirror reflection in X ,X terms. After such transformations the zero degree
0
Poisson structure looks as
0 σ
−σ 0
(cid:18) (cid:19)
where σ is matrix (1+N1+N0×1+N1+N0) with different from zero unity
on its main anti diagonal.
The proof of conservation of the Poisson brackets may be obtained more
directly from the fact of existence of Lagrangian function [14] ( and action for
n-wave system)
1 1
L(f)=Sp ([d,f ]+[c,f ])+ (f[[d,f][c,f]]),S = dtdxL(t,x)
t x
2 3
Z
change on the derivatives under transformation of maximal root above.
4.1 Frechet derivative
TheFrechetderivativeof(5)inconnectionofitsdefinition(2)maybepresented
in block form as 5×5 matrix valued operator
0 0 0 0 φ′
M,−M
0 0 0 φ′ φ′
 00 φ′ 0 φφ′′r0,r0′ φ′φR′r(+0,1S,S(−(−11 φφR′′r(+0,1−,−MM (9)
φ′M,−M φS′−(M−1,,RR++11 φS′−(M−1,,rr00 Sφ(′−−M1,(,SS((−−11)′ φS′−(M−1,,−−MM 
 
where I is the unity matrix with the dimension of zero grading subspace: the
notationM means that fromthe algebravalued function M necessaryextract
R
coefficientundercorrespondingrootRofthealgebrabelongingtogradingspaces
with indexes ±2,±1 and 0.
φ′ =− 1 , φ′ = ([XM+,XS(−1]X−R(+1),
M,−M (f−)2 R(+1,S(−1 f−
M M
φ′R(+1,−M =−([XM+,f((f−−1])X2−R(+1), φ′r0,r0′ =I, φ′r0,S(−1 = ([[XM+,XS(−f1−]f(−1]X−r0)
M M
7
1([[X+,f(−1]f(−1]X ) f−
φ′r0,−M =−2 M (fM−)2 r0 , φ′S(−1,R+1 = cSM(−1([[c,XR+1]XM−]X−S(−1),X
−
1 f
φ′S(−1,r0 =−cS(−1([f(−1[c,Xr0]]X−S(−1), φ′−M,M =(fM−)2, φ′−M,R(+1 =−cMM(f(−1[c,XR(+1])
φ′−M,r0 = 2c1 ([[XM+,f(−1]f(−1][c,Xr0]), φ′−M,S(−1 = cSc2(−1([XM+,f(−1]XS(−1)D+
M M
−
1 f 1
6cMfM−([XM+,f(−1]XS(−1][[XM+,f(−1]f(−1])+cMM([c,f(+1]XS(−1)−cM([[XM+,XS(−1]f(−1][c,f0])−
1 (f−)′− 1([X+,f(−1]([c,f(−1])
(c )2([[XM+,XS(−1][c,(f(−1)′])+(2cS(−1+cM) M 2 cM2 f− ([XM+,f(−1]XS(−1)
M M M
φ′S(−1,−M = ff−S(c−1 D−(ff−S)(−2c1 (fM−)′−6(f1−)2(X−S(−1[[[XM+,f(−1]f(−1]f(−1])+
M M M M M
2cMfS((f−M−1 )2c1M)([XM+,f(−1][c,f(−1])+ cS1(−1([XM−,X−S(−1][c,f(+1])
φ′S(−1,(S(−1)′ =δS(−1,(S(−1)′(cS1(−1D+(fM−cM)′f−M−12Θ)−c(S(c−M1)f′fM−S(−1(X(S(−1)′[XM+,f(−1])−
1 1
cS(−1(X−S(−1[X(S(−1)′[c,f0]])+ 2fM−(X−S(−1[[α(+1,f(−1]X(S(−1)′]
1 ((f−)′− 1Θ) ((f−)′− 1Θ)2
φ′ = D2−2 M 2 D+2(f−f+ + M 2 −
−M,−M (cM)2 (cM)2fM− M M (cM)2(fM−)2
1 1
([α(+1,f(−1][α(+1,f(−1])− (f(−1[c,f(+1])
24(fM−)2 cM
5 The main assertion
TherearetwoPoissonstructuresJ andJ ,whichareinvariantwithrespectto
0 1
discrete transformation of the maximal root of the previous subsection (equa-
tions (4) are satisfied) with the explicit form of its matrix elements
1
<R|J0|R′ >=δR,−R′c ,c−R =−cR (10)
R
The fact of invariance of J Poissonstructure with respect to discrete transfor-
0
mation was proofed in the previous section.
1 1 1
<R|J1|R′ >=−fR(hR,hR′)D−1fR′ + cR[(X−R,X−R′][c,f])cR′ + c2RδR,−R′D
(11)
whereh = nRh ,nR multiplicitysimpleroot±αintherootRandh Car-
R α α α α
tanelementofthissimpleroot. Theproofofthelast(notsotrivial)proposition
P
will be presented below.
8
5.1 Simplest nontrivial solution of (3)
The column function F =g f is the solution of the equation (3). We have
R R R
(φ′(f)F)|R>= <R|φ′(f)|R′ >FR′ = <R|φ′(f)|R′ >gR′fR′ =
X X
∂∂TffMRR→′ [g,fR′]=[g,TfMR→]=gR TfMR→
X
Indeed g f may be considered as result of differentiation [g,f ] and the dis-
R R R
crete transformation (5) conserve the grading structure with respect to simple
roots of the algebra.
5.2 Equation which it is necessary to check
Let us rewrite the equation for Poisson structure J (4) taking into account the
fact of existence of J (10). We have (JT =−J)
0
φ′(f)J =TMJ→(φ′(f)−1)T =−(φ′(f)−1 TMJ→)T =−(J φ′(f)TJ−1 TMJ→)T (12)
0 0
Let us present first term in (11) in equivalent form
r
fR(hR,hR′)D−1fR′ = FiD−1FiT
i=1
X
where F = gi f are r independent nontrivial solutions of the first subsec-
i R R
tion. Then the form above is equivalent to (11) after identification ((gR,gR′)≡
igRi gRi′ =(hR,hR′). And this fact it will necessaryto prove. Now let us con-
sider result of multiplication Frechet derivative on (column) nontrivial solution
P
F . Using the rule of multiplication of quadratical in derivatives operator on
i
scalar function F
(A+BD+CD2)=AF +BF′+CF′′+(BF +2CF′)D+CFD2 (13)
and explicit form of Frechet derivative we conclude that first 1+n1+n0 lines
ofitdoesnotcontainoperatorofdifferentiationD,nextn1 onesarelinearinD
and the last one is quadratical in it (we remind that 1-dimension of ±2 graded
subspaces, n1,n0 dimensions of ±1,0 ones). We present below n1+1×n1+1
matrix in the left dawn corner of Frechet derivative which contain operator of
differentiation in explicit form
cS1−1δ(S−1)′,S−1D cfMS−fM−1 D
c12 ([c,X(S−1)′][XM+,f(−1])D c12 D2−2((fM−)′−21([cc,f2(−f−1][XM+,f(−1]))D
M M M M
 (14)
9
TM→
In connection with (3) F = φ(f)F (in the last formulae all operators of
i i
differentiation acts directly on F ). And thus we have
i
0
1+n1+n0
φ(f)Fi =TMFi→+ −(gcMMi − gcSSi−−11)fS−1D ≡ (15)
−gMic2MfM−D2+ c12M([c,f(−1][XM+,[gi,f(−1]])D
 
TM→
F +Γ
i i
The assumed form of second Poisson structure contain terms with negative,
zero and positive degrees of D (J−1,J0,J11). The (15) together with (12) and
1 1
comments after leads to conclusion
φ′(f)J−1 =− r (TMF→D−1FT +Γ D−1FT)
1 i i i i
i=1
X
But Γ is proportional to D and thus last term in the sum above contain only
i
0,1 degrees of D. Thus explicit expression for − r (Γ D−1FT) looks as
i=1 i i
0 P
1+n1+n0
 ((McM,P) − (Sc−S1−,1P))fS−1 (FP)T
(M,cP2)fM−D− c12 ([c,f(−1][XM+,[ (gigPi ),f(−1]])
 M M 
 
P
Inthelastcolumndifferentfromzeroonlycomponentsofitsn +1lastrows. In
1
whatfollowsweusenotationΘ=([c,f(−1][X+,f(−1]]),Θ =([c,f(−1][X+,[ (gigi ),f(−1]]).
M P M P
Now let us inverse (15) presenting it in a form
P
FT =(TMF→)T(φ−1(f))T +ΓT(φ−1(f))T
i i i
Forcalculation(φ′(f)−1)T =J−1φ′(f)J itissuitabletopresentJ asprod-
0 0 0
uctoftwomatricesonediagonalonewithmatrixelements<R|d|R′ >=δR,R′c1R
and the second with different from zero unites on its main anti diagonal <
R|ad|R′ >=δR,−R′. Thus we obtain (φ′(f)−1)T =J0−1φ′(f)J0=
φ′ φ′ φ′ φ′ φ′
−M,−M −M,−(R(+1)′ −M,−r′ −M,−(S(−1)′ −M,M
φ′ φ′ φ′ 0 φ′ 0
 −R(+1,−M −R(+1,−(R(+1)′ −R(+1,−r′ −R(+1,−(S(−1)′ 
d−1 φ′ φ′ I 0 0 0 d
φ′−−Sr(−0,1−,−MM φ′−−Sr(−0,1(,−−R(R(+(+1)1′)′ 0 0 0 
 φ′ 0 0 0 0 
 M,−M 
 (16)
10