A Lie algebraic generalization of the Mumford system, its symmetries and its multi-Hamiltonian structure,

its symmetries and its multi-Hamiltonian struture

Maro Pedroni 1

and Pol Vanhaeke 2;3

Abstrat

In this paper we generalize the Mumford system whih desribes for any xed g all linear

ows on all hyperellipti Jaobians of dimension g. The phase spae of the Mumford system

onsists of triples of polynomials, subjet to ertain degree onstraints, and is naturally seen as

an aÆne subspae of the loop algebra of sl(2). In our generalizations to an arbitrary simple Lie

algebragthephasespaeonsistsofdimgpolynomials,againsubjettoertaindegreeonstraints.

This phase spae and its multi-Hamiltonian strutureis obtained by a Poisson redution along a

subvarietyN of theloop algebra g((

1

))of g. Sine N is nota Poissonsubvariety forthewhole

multi-Hamiltonianstrutureweprovean(algebrai)Poissonredutiontheoremforredutionalong

arbitrarysubvarietiesofan aÆnePoissonvariety; thistheoremis similarinspiritto theMarsden-

Ratiu redutiontheorem.

Wealsogiveadierentperspetiveonthemulti-HamiltonianstrutureoftheMumfordsystem

(anditsgeneralizations)byintroduingamastersymmetry;thismastersymmetryanbedesribed

on the loop algebra g((

1

)) as the derivative in the diretion of and is shown to survive the

Poissonredution. Whenating(asaLiederivative)ononeofthePoissonstruturesofthesystem

it produes a next one, similarlywhen atingon one of theHamiltonians (in involution) ortheir

(ommuting)vetoreldsitproduesanextone. Inthiswaywearriveatseveralmulti-Hamiltonian

hierarhies,builtup by amaster symmetry.

Mathematis Subjet Classiation (1991): 58F07,17B81.

Key words: Mumford system, simple Lie algebras, multi-Hamiltonian strutures, Poisson

redution,master symmetry.

1

Universita diGenova, Dipartimento diMatematia, ViaDodeaneso 35,I-16146 Genova, Italy

(pedronidima.unige.it)

2

UniversityofCalifornia, 1015 Department of Mathematis,DavisCA 95616-8633, USA

(Vanhaekmath.udavis.edu)

3

Universite desSienes etTehnologiesde Lille,U.F.R. de Mathematiques, 59655 Villeneuve

d'AsqCedex, Frane

1 Introdution

2 Multi-Hamiltonianhierarhiesand master symmetrieson loop algebras

2.1 Theloop algebra

~

g=g((

1

))and its Poisson brakets

2.2 Mastersymmetriesandthe deformationproperty

3 Poisson redutionandredutionof symmetries

4 RedutionforsimpleLie algebras

4.1 Poisson redution

4.2 Redutionof themastersymmetry

4.3 Thereduedspae N=G

asa linearsubspaeN

0 N

5 Examples

5.1 TheMumfordsystem

5.2 A

r

5.3 B

r andC

r

5.4 D

r

5.5 G

2

In its original form the Mumfordsystem onsists, for every positive integer g, of a family of

vetor eldson an aÆne spae (of dimension 3g+1) of triplesof polynomials(U();V();W())

(see [Mum℄p.3.43). The simplestmemberofthisfamilyhasthe form

_

U()=V();

_

V()= 1

2

[ W()+( U

g 1 +W

g

)U()℄;

_

W()= ( U

g 1 +W

g )V():

(1:1)

The three polynomials are subjet to the restritions that U and W are moni of degrees g and

g+1, whileV has degree less than g. U

i

is theoeÆient of i

inU() and similarly forV and

W. Asimpleomputation shows that

(U()W()+V 2

())

=0

fortheabove vetor eld,and similarlyfortheother membersof thefamily. It follows thatif one

assoiatesan algebraiurve(of genusg) to everypoint (U();V();W()) of phasespae bythe

equation

2

=U()W()+V 2

(); (1:2)

then thisurve is invariant under theow of these vetor elds. This property is \explained" by

Mumfordwhoshowsthatthegeneriorbit,traedoutbytheowofthesevetorelds,isanaÆne

partof theJaobianoftheurve (1.2) assoiatedtoanyof itspoints(U();V();W()) and that

theows of these vetor eldsare linear (the Jaobianof aurve is a omplex torus,hene has a

linearstruture). Notethatthisimpliesautomatiallythatthesevetoreldsommute,aproperty

reminisentof integrablesystems. UponintroduingaHamiltonianstrutureforwhihMumford's

vetor elds are Hamiltonian it turns out that the Mumford system is indeed an example of an

integrable system(suh aHamiltonianstruturewashowever onlyintroduedlater).

ItturnsoutthattheMumfordsystemandsome ofitsgeneralizationsappearinmanydierent

ontexts, although sometimes in a disguised form and often without referene to its Hamiltonian

struture. Itappears inthedesriptionof ringsofommuting dierentialoperators,goingbakto

the early papers of Burhnall and Chaundy (see [BC℄ or [Pre℄; for a dierent but equivalent de-

sriptionsee[Sh℄),itisalimitofthelassialShlesingerequationswhihdesribeisomonodromy

deformation (see [Gar℄), many lassial integrable systems are isomorphi to a subsystem of the

Mumford system, sometimes up to a over (see [Van3℄) and the Mumford system appears as the

simplestofa largelassof integrable systemson themodulispaeof Higgsbundleson aRiemann

surfae, thelatterbeing inthisasejustthe Riemannsphere(see [DM℄).

Thepurposeofthispaperistoombine theideasin[MM℄,[MR℄,[RS3℄and[Sh℄togeneralize

the Mumford system and to desribe the symmetries and the multi-Hamiltonian struture of its

generalizations. Let usdesribethese ideas.

(1) The mainidea from [RS3℄, whih is realled inSetion 2.1, is that the loop algebra

~

g =

g((

1

))ofanysemi-simpleLiealgebrahasa(multi-)Hamiltonianstruturewhihrestritstothe

nite-dimensionalspaes

~

g

n

of polynomials withleading term n

, where 2 g. A natural lass

of funtionsininvolutionleads, inmanyases, to anintegrablesystemon

~

g

n

=G

where G

isthe

isotropygroup of. Wewillshowthatforwell-hosen aPoissonredutiononan aÆnesubspae

N of

~

g

n

withrespettoasubgroupG

ofG

willleadtothegeneralizationoftheMumfordsystem:

while

~

g

n

=G

isneveranaÆnespaethequotientwhihwedesribewillbeaspaeof(dimg)-tuples

of polynomials withdegree onstraints, preiselyasinthease ofthe Mumfordsystem.

(2)The multi-Hamiltonianstruture on

~

g

n

,whihis given as a familyof ompatible Poisson

brakets,doesnotrestrittoanysubvarietyof

~

g

n

(althoughsomebraketsdo). Thereforeweprove

a general (algebrai) Poisson redution theorem whih is similar in spirit to the Marsden-Ratiu

redutiontheorem (see [MR℄). Inour theorem we onsidera subvarietyN of a Poisson varietyM

onwhihanaÆnePoissongroupGats(leavingN invariant). AssumingthattheationisPoisson

wegive aneessaryandsuÆient onditionforthePoissonstrutureonM to desendtoaPoisson

strutureon N=G. The redutiontheoremwillbe proven inSetion3.

(3)The next question then, whih turnsoutto be Lie-algebrai innature, is howto pikthe

subspae N and the group G

suh that the quotient is an aÆne spae whih an be naturally

identiedwith a subspaeof N. If we pikin thease of sl(r+1) theleading oeÆient to be

a generi lowertriangular matrix, then ourondition forPoisson reduibilityimpliesthat we an

onlyredue alongthehyperplaneN

~

g

n

whihisobtained byxingone of theentries(the entry

at position(1;r+1))of the oeÆient of n 1

. In thisase thequotient N=G

is an aÆne spae

ifandonlyifthisentry hasbeenxedto a valuedierent from0. Notiethatinthisase+ is

regular,where

= 0

B

B

0 ::: 0 1

0 ::: 0 0

.

.

. .

.

. .

.

. .

.

.

0 ::: 0 0 1

C

C

A :

The preise Lie-algebrai ondition imposedon is that itis a prinipal nilpotent element; sine

all prinipal nilpotent elements of a simple Lie algebra are onjugate we may take to be given

by = P

r

i=1 F

i

,where fH

i

;E

i

;F

i g

i=1;:::;r

is a Weyl basisof g . Then theondition whih denes

N

~

g

n

is thatits elements n

+ P

n 1

i=0 x

i

i

satisfy

k x

n 1

=; is anynon-zero top-element

in the gradation k

i= k g

i

of g , whih is assoiated to the Weyl basis, i.e., 0 6= 2g

k

and

k is

theprojetionontog

k

. ThisleadstotheproperLiealgebraisetupfora rstgeneralizationofthe

Mumfordsystem to any simpleLie algebra. Indeed, ifwe dene; and n inthe above wayfor

anarbitrarysimpleLiealgebragthenthewholemulti-HamiltonianhierarhyofPoissonstrutures

andthealgebraoffuntionsininvolutiondesendto thequotientwhihisnaturallyidentiedwith

an aÆne subspae N

0 of

~

g

n

. Notie that the group by whih we redue is in this ase the full

group G

, whih is Abelian, and that the ation is Hamiltonian. In the ase of sl(2) we reover

theMumfordsystem, whileintheaseofsl(r+1)wendageneralizationoftheMumfordsystem

dueto Donagi-Markman (see[DM℄).

(4)When isnotaprinipalnilpotentelementthenthewhole struturetheoryofsimpleLie

algebras omes into play. Indeed, we will rely heavily on the beautiful paper [Kos1℄ by Kostant.

As we learned from A. Shwarz, forany d oprimeto r+1, thespae of matries in sl(r+1) of

theform

0 0

I

r+1 d 0

n

+

? M

d

? ?

n 1

+ n 2

X

i=0 x

i

i

; (1:3)

where M

d

is any lower triangular matrix of size d with ones on the diagonal, appears in the

desriptionofthesolutionstothestringequation[P;Q℄=1,or,inananalogousway,ofthesolutions

to the ommutativity equation [P;Q℄ = 0; in these equations P and Q are dierential operators

subjet to ertain normalizations (see [Sh℄ and [KV℄). Notie that the matrix

0 I

d

I

r+1 d 0

,

whih is obtained from the leading oeÆients, is regular due to the fat that d and r+1 are

nd that the multi-Hamiltonian hierarhy redues and that the quotient spae an be naturally

identiedwith an aÆne subspae of N. A key informationwhih we also learned from Shwarz'

desriptionisthatwe shouldnotatwiththefullisotropygroupG

butwiththesubgroupG

of

lowertriangularmatriesinG

(withonesonthediagonal). InSetion4weimplementthese ideas

intheaseofan arbitrarysimpleLiealgebragand ndforanyhomogeneous theorresponding

subspaesN of

~

g

n

towhihthemulti-Hamiltonianhierarhiesredue;moreoverwegivethehoies

of whih leadto a quotient whihisaÆne,therebygiving theLie algebraiinterpretation ofthe

oprimeonditionwhihappears inthease ofsl(r+1). Notiethatinthismore generalaseG

isnot Abelian; moreover itan beshown thattheation isnotHamiltonian.

(5)Anotheridea,whih welearnedfrom[MM℄, isthatmulti-Hamiltonianstruturesareoften

built up from a basione byapplying a master symmetry, i.e., there is a basi Poisson struture

whosesuessive Lie derivatives withrespetto a ertain vetor eld V provides alinear basisfor

all thePoissonstrutures. Thisvetor eldV isgiven on

~

g

n by

_

X()=

X();

and it generates all theHamiltonians and ommuting vetor elds startingfrom afew basi ones.

Weshowthat,asaonsequeneofourredutiontheorem,thevetoreldV projetsonthequotient

to amaster symmetry whih buildsupthe multi-Hamiltonianstrutureon thequotientspae N

0 .

Sine theoperations ofredution and takingthe Lie derivative ommute itfollows that it suÆes

to omputethe redutionof one Poisson strutureon

~

g

n

(infat preisely theunique linear one)

and apply suessive Lie derivatives to it to ndthe other reduedPoisson strutures. The basi

properties ofthevetor eldV willbegiveninSetion2.2,wedisusstheredutionofsymmetries

at theendof Setion3 and we ndthe redued master symmetry inthe ase of the loop algebra

inSetion4.2.

We will endthis paper with a list of examples (Setion 5). We will rst show how our on-

strution speializes in the ase of sl(2) to the Mumford system. In this ase we will expliitly

omputeallreduedbrakets. Wewillgiveanexpliitdesriptionofthequotient spae(asaspae

of polynomials)forthelassialLie algebrasand forG

2 .

Inonlusionwe havea ompletedesriptionofthe multi-HamiltonianstrutureoftheMum-

fordsystemandits generalizationstoarbitrary simpleLie algebras. Itseemsnon-trivial butinter-

estingto dothe samefortheeven master system(see [Van1℄),whihalso desribesall linearows

on all hyperelliptiJaobians byequations whih aresimilarto (1.1). A proof of theintegrability

ofthesystemsonthereduedspaeN

0

involvesalgebraigeometriarguments,revealingalsotheir

algebrai omplete integrability (this is done for the ase of sl(n) in [DM℄); we leave this and a

study of the algebrai geometry of the bers of the Hamiltonians | some of whih are ertainly

interesting Abelianvarieties | forthefuture.

Aknowledgments.We aregratefultomanypeoplefortheirinterestinthispaperandforuseful

related disussions. Among them we wish to mention Mark Adler, Franeso Bottain, Paolo

Casati, Filippo De Mari, Laszlo Feher, Hermann Flashka, Guido Magnano, Motohio Mulase,

SergeParmentier,AlexeiReyman,AlbertShwarz,MihaelSemenov-Tian-ShanskyandPierrevan

Moerbeke. Speial thanks are due to Frano Magri, who started this projet with us, and to

BeltramKostantwho provideduswith aproof ofProposition4.8. Thesupportof MURSTand of

theGNFMoftheItalianCNRisalsogreatly aknowledged. TherstauthorthankstheUniversite

Catholiquede Louvain andthe Universityof Californiaat Davisfortheirhospitality.

algebras

In this setion we introdue a large lass of multi-Hamiltonian hierarhies on the loop alge-

bra

~

g=g((

1

)),wheregisanite-dimensionalLiealgebra,whihisequippedwithanad-invariant

non-degenerateinnerprodut. Themulti-Hamiltonianstrutureofinterestherewasrstintrodued

in [RS3℄ by using several R -brakets and is realled in Paragraph 2.1. We exhibit several multi-

Hamiltonianhierarhies,whoseHamiltoniansareseento beininvolutionbythelassialR -matrix

argument; we providean alternative proofwhih uses thelassialLenard-Magri sheme. Follow-

ing an idea of[MM℄ we showin Paragraph 2.2 thatthe dierent Poisson brakets whih make up

the multi-Hamiltonianstruture are onnetedby theLie derivative along a master symmetry V,

thereby giving another, more geometri, onstrution of these brakets. The vetor eld V allows

one topassfrom oneHamiltonian(anditsvetor eldwithrespetto anyofthePoissonbrakets)

to another, hene playing a similarrole asthe reursionoperatorin thease of Poisson-Nijenhuis

manifolds(see [KM℄).

2.1. The loop algebra

~

g=g((

1

)) and its Poisson brakets

Let g be a (nite-dimensional) Lie algebra and h; i

g

a non-degenerate inner produt whih

is ad-invariant, h x;[y;z℄i

g

= h [x;y℄;zi

g

. We x a basis fe

a g

a2I

for g and dene linear forms

a

:g!C by

a

=h;e

a i

g

. We lookat g asan aÆne (algebrai) variety, inpartiularwe onsider

O(g)= C[

a

℄

a2I

as its algebra of (regular) funtions. For any F 2 O(g), its gradient rF(x) at

x2g isdened by

hrF(x);yi

g

= d

dt

jt=0

F(x+ty) 8y2g:

For anya2I and F 2O(g)the map x7! h rF(x);e

a i

g

belongsto O(g). It follows that for any

F;G2O(g)thePoissonbraketfF;Gg,dened by

fF;Gg(x)=h x;[rF(x);rG(x)℄ i

g

; (2:1)

also belongs to O(g), making O(g)intoa Poissonalgebra.

From g we onstrut the loop algebra

~

g = g((

1

)) = g[℄

1

g[[

1

℄℄: Elements of the

loop algebra willbe denoted by apital letters; for an element X =X() = P

x

i

i

2

~

g we write

X=X

+

+X aordingto the above (vetor spae) deomposition. The innerprodut h;i

g on

gleadsto an innerproduth ;i on

~

gvia

hX();Y()i= X

i+j= 1 h x

i

;y

j i

g :

By a slight abuse of notation one often writes ReshX();Y()i

g

for the above right hand side;

here Res P

x

i

i

= x

1

. Clearly h;i is ad-invariant and non-degenerate just as h;i

g

is. For

a2I; i2Z we dene elements E i

a

=e

a

i

of

~

gand linearfuntions i

a

=

;E i 1

a

. We wishto

introdueanalgebraO(

~

g )offuntionson

~

gforwhihweandeneagradientandaPoissonbraket

asintheaseofg,butwhihislargeenoughtoontainfuntionsofthetypeX()7!ResH(X())

(for H 2O(g)), whih willbe important later. To do thiswe rst dene on

~

g

n

= n

g[[

1

℄℄ an

algebraof funtions by

O(~

g

n )=C

i

a

a2I

in

and obtainfromit thefollowingalgebraof funtions ong:

O(

~

g )= n

F :

~

g!Cj8n2Z:F

j

~

g

n 2O(

~

g

n )

o

:

Thus, elements of O(~

g ) restrit to polynomials on all subspaes

~

g

n

. As in the ase of g the

gradient rF(X) ofa funtionF 2O(

~

g )at X 2

~

g isdened by

hrF(X);Yi= d

dt

jt=0

F(X+tY) 8Y 2

~

g: (2:2)

Proposition 2.1 For any X 2

~

g and F 2 O(

~

g ), rF(X) is well-dened by (2.2) and belongs

to

~

g. For any F;G2 O(~

g) the Poisson braket fF;Gg, dened by

fF;Gg(X)=hX;[ rF(X);rG(X)℄i;

belongs to O(

~

g), making O(

~

g) into a Poisson algebra.

Proof

Thefatthat thegradientis well-denedfollowsfrom non-degeneray ofh;i; infat, forany

j2ZtheoeÆient (rF(X))

j

2gis givenby

h(rF(X))

j

;e

a i

g

=

rF(X);E j 1

a

= d

dt

jt=0

F(X+tE j 1

a ):

IfX 2

~

g

n

thenF(X+tE j

a

)isindependentoftforjsuÆientlysmall,sineF

j

~

g

n

isapolynomial.

Thus, (rF(X))

j

is zero for j suÆiently large and rF(X) 2

~

g. Further, X 7!

rF(X);E j

a

belongsto O(

~

g ) for anyj 2Zsinetherestrition to any

~

g

n

of themap

X 7!

d

dt

jt=0

F(X+tE j

a )

is just a polynomial (inthis formula, useF

j

~

g

m

where m =maxfn;jg). As a orollary, ifF;G2

O(

~

g) thenthemap

X7!hX;[ rF(X);rG(X)℄i

belongsto O(

~

g ),givinga braketf;g:O(

~

g )O(

~

g )!O(

~

g ). The fatthat itsatisestheJaobi

identityfollowsfrom thefatthat (2.1) satisestheJaobiidentity.

Following[RS3℄ weintroduea familyR

l

ofendomorphismsof

~

gby

R:

~

g!

~

g:X7! X

+ X ;

R

l :

~

g!

~

g:X7! R ( l

X):

Proposition 2.2([RS3℄) For any l2Za Poisson braket on O(~

g) is dened by

fF;Gg

l (X)=

1

2 hX;[ R

l

rF(X);rG(X)℄+[ rF(X);R

l

rG(X)℄ i:

l

ombinationof these brakets isa Poisson braket.

Asabove thesebraketsaretakenasbraketson O(~

g ). WeallthemR -braketsandallf;g

the anonial Lie-Poisson braketon

~

g. If we denote thestruture onstants of g with respetto

the basisfe

a g

a2I by C

ab

, i.e., [e

a

;e

b

℄= P

2I C

ab e

, then one easily nds by usingr i

a

= E i 1

a

that

f i

a

; j

b g

l

= ij

l X

2I C

ab

i+j+1 l

; (2:3)

where ij

l

= 1 if i;j < l and ij

l

= 1 if i;j l; otherwise ij

l

= 0. The R -brakets have two

remarkable properties whih make them more relevant for integrable systems than the anonial

Lie-Poisson braket on

~

g. The rst property, whih follows immediately from (2.3), is that if

plq+1then f;g

l

restritsto thefollowingnaturalnite-dimensionalsubspaeof

~

g,

~

g

p;q

= 8

<

: q

X

i= p x

i

i

jx

i 2g

9

=

;

: (2:4)

Sine multipliation by p

indues an isomorphism

~

g

p;q

;f;g

l

!

~

g

0;p+q

;f;g

l+p

we may

restrit ourselves to thespaes

~

g

0;n

of matries whih are polynomial (in )of degree at most n.

In fatwe willbe interested intheaÆne subspaes of

~

g

0;n

dened by

~

g

n

= (

n

X

i=0 x

i

i

2

~

gjx

n

= )

; (2:5)

where isanyxedelementing. ThefamilyofR -braketswhihrestritsto

~

g

n

isalso omputed

at onefrom (2.3) and isgiven inthefollowingproposition.

Proposition 2.3 If is not a entral element in g then the Poisson struture P

1

l= 1

l f;g

l

restrits to

~

g

n

ifand only if

l

=0 for l<0 and for l>n.

TheseondremarkablepropertyoftheR -braketsisthattheAd-invariantfuntionsonglead

to a large subalgebra A of O(~

g) whih is involutive with respet to all these brakets. Indeed,

a funtion H 2 O(g) indues a funtion H :

~

g ! C((

1

)) and hene leads for any i 2 Z to a

funtionH

i on

~

g ,dened by

H

i

(X())=Res

H(X())

i+1

: (2:6)

Clearlyanysuh funtionH

i

belongsto O(

~

g ).

Proposition 2.4 ([RS3℄) Let H and K betwo Ad-invariant funtions in O(g). Then for any

i; j2Z the funtions H

i

and K

j

arein involution with respet to all R -brakets f;g

l .

Proof

Ad-invariantfuntionsinO(g )arethosefuntionswhihareinvariantfortheadjointationof

aLiegroupGforwhihg=LieG. Itmaybeimpossible 1

topikGalgebraibutthisis irrelevant

1

Ifg issemi-simplethen Gisalgebrai, see [OV℄, p.29.

[x;rH(x)℄=0. Toshow thelatter,usead-invarianeof h; i

g

and Ad-invarianeof H to nd

h[ x;rH(x)℄;yi=h rH(x);[y;x℄i= d

dt

jt=0

H(x+t[y;x℄)= d

dt

jt=0 H(Ad

z(t)

x)=0;

when setting z(t) = expty. In partiular, if H 2 O(g) is Ad-invariant then, for any i 2 Z, the

funtionH

i 2

O(~

g)denedby(2.6) isAd-invariant and[X;rH

i

(X)℄=0. It followsthatifH and

K areAd-invariantfuntions on gthenforanyi;j;l2Z

fH

i

;K

j g

l (X)=

1

2 hX;[ R

l rH

i

(X);rK

j

(X)℄+[rH

i (X);R

l rK

j

(X)℄ i=0;

showingthat allfuntions on O(~

g ) whihome fromAd-invariant funtionson gare ininvolution

withrespetto all R -brakets.

The algebra of Ad-invariant funtions on g is denoted by O(g) G

and the involutive algebra

generated by all H

i

;i 2Z; H 2 O(g) G

is denoted byA. If we dene for any F 2 O(~

g) a vetor

eldon

~

gbyX

F

=f;Fg

0

thenthei-thvetor eldX

Hi

(i2Z)whihomesfromanAd-invariant

funtionH 2O(g ) G

isgiven bytheLaxequation

_

X= 1

2

[X;R rH

i

(X)℄: (2:7)

Two alternative waysto writethisare

_

X= [X;( rH

i (X))

+

℄=[X;( rH

i

(X)) ℄: (2:8)

Thevetor eldsX

H

i

areinfatHamiltonianwithrespetto allbrakets f;g

l

. Toseethis,hek

thatforanyH 2O(g),

d

dt

jt=0 Res

H(X+tY)

i+1

= d

dt

jt=0 Res

H(X+tY)

i+2

;

showing that rH

i

(X) = rH

i+1

(X). It follows that (2.7) an be written in Lax form with

respettoall endomorphismsR

l

andthatforanyH2O(g ) G

thefuntionsfH

i g

i2Z

forma multi-

Hamiltonianhierarhyinthesensethat

f;H

i g

0

=f;H

i+l g

l

(i;l2Z): (2:9)

The relations (2.9), whihare alled Lenard relations, an be used to give an alternative proof of

Proposition2.4. Forfuntions belongingto thesamehierarhythelassialargument applies(see,

e.g., [CMP℄),giving fH

i

;H

j g

l

=fH

j

;H

i g

l

=0:Formembersof dierenthierarhies,oming from

dierent funtions H;K 2 O(g ) G

some are is needed sine none of the H

i or K

j

is a Casimir

foranyof the R -brakets. However, we see from (2.8) that foranyX 2

~

g theHamiltonianvetor

eld X

H

s

vanishesat X fors largeenough sinethen (rH

s (X))

+

=0. Thusalso inthisase the

Lenard relationsgive (e.g., forthezeroth R -braket)

fH

i

;K

j g

0

(X)=fH

s

;K

j s+i g

0

(X)=0:

whih shows thatfuntions whih belong to dierent hierarhiesarealso ininvolution.

InthisparagraphweshowhowthePoissonbraketsf;g

l

arerelatedbyavetoreldV whih

is a master symmetry 2

for the involutive algebra A (introdued after Proposition2.4). We mean

bythis that V has theproperty [[ V;X

F

℄;X

G

℄=0 forall F;G 2A (a symmetry has the stronger

property[V;X

F

℄=0 forall F 2A). The vetor eld V has inaddition the deformation property

withrespet to thebraketsf;g

l

;thismeansthat theLie derivativeof anybraket f;g

l inthe

diretionofV isalsoaPoissonbraket 3

. Aswasshownin[MM℄thisimpliesthatanybraketf;g

l

isompatible withits Lie derivative inthediretionof V.

The vetor eld V is dened as the innitesimal generator of the ation of C on

~

g given by

\shiftin",

s;

X

x

i

i

7!

X

x

i

(+s) i

;

herewe usefornegativepowers ofthe formalexpansion

(+s) 1

= X

i0 ( 1)

i

s i

i 1

;

whih is atually onvergent for small s, in partiular it is the right denition if one wants to

onsiderthefundamentalvetor eldV of thisation: thelatter iseasilyomputed as

_

X()=

X() i.e. L

V

j

a

=(j+1) j+1

a

;

whereL

V

denotestheLie derivative alongV. The two mentionedpropertiesof V aregiven bythe

followingproposition.

Proposition 2.5 Let i;l2Z and H2O(g) G

be arbitrary.

a) V has the deformation property with respet to all brakets f;g

l

, more preisely the

relation

L

V fF;Gg

l fL

V F;Gg

l

fF;L

V Gg

l

= lfF;Gg

l 1

(2:10)

holds, i.e., the Lie derivative of the l-th R -braket is (up to a fator l) the (l 1)-th

R -braket;

b) L

V H

i

=(i+1)H

i+1

;

) [V;X

H

i

℄=X

L

V H

i

=(i+1)X

H

i+1

;

d) V isa master symmetry for A.

Proof

It suÆes to verify a) for F = i

a

and G= j

b

with sayij. We an use (2.3); sinefor this

partiular F and G all terms in (2.10) are proportional to P

C

ab

i+j l+2

it atually suÆes to

keep trak of the oeÆients and the proof of (2.10) amounts to the veriation of the following

identity,

(i+j l+2) ij

l

(i+1) i+1;j

l

(j+1) i;j+1

l

= l

ij

l 1 :

2

TheoneptofamastersymmetrywasrstintroduedbyFuhssteiner(see[Fu℄). Thenotion

we usehere isslightlymore general.

3

In many important examples the master symmetries for an algebra whih is involutive with

respettosomePoissonbrakethavethedeformationpropertywithrespettothisPoissonbraket,

however these two properties areindependent ingeneral.

L

V H

i (X)=

d

dsjs=0 Res

H(X(+s))

i+1

=Res 1

i+1

d

d

H(X())

=Res

d

d

H(X())

i+1

+(i+1)

H(X())

i+2

=(i+1)Res

H(X())

i+2

;

whih ispreisely(i+1)H

i+1

(X). For)wesubstitute l=0 andG=H

i

inparta) to nd

L

V fF;H

i g

0

=fL

V F;H

i g

0

+fF;L

V H

i g

0

;

whih an also be written as L

V (X

H

i

(F))=X

H

i (L

V

F)+X

L

V H

i

(F); usingb) we onlude ). In

orderto showd) rstnotiethat[X

F

;X

G

℄= X

fF;Gg

0

=0foranyF;G2A. Then)impliesthat

[[V;X

H

i

℄;X

G

℄=0foranyH 2O(g) G

and foranyG2A. BytheJaobiidentitywealso havethat

[[V;X

G

℄;X

H

i

℄ = 0: The more general statement that [[V;X

G

℄;X

F

℄ = 0 for any F;G 2 A follows

from b) upon usingthe fat that A is generated by the funtions H

i

where i runs over Z and H

runs overO(g ) G

.

Pikingany two Poissonstrutures suh asf;g

0

and f;g

l

the relations(2.9) and Proposi-

tion 2.5an bedepited inthefollowingdiagram (we omitthe oeÆients; L l

V :=L

V ÆL

l 1

V ),

:::

H

i L

l

V

! H

i+l L

l

V

! H

i+2l

0

?

?

y

.l 0

?

?

y

.l 0

?

?

y

X

H

i L

l

V

! X

H

i+l L

l

V

! X

H

i+2l

:::

Remark 2.6 An R -braket on a Lie algebra g leads also to a quadrati and a ubi braket,

assuming that the Lie algebra derives from an assoiative algebra, with a pairing h ;i

g whih

derivesfromatraeform(see[LP℄and[OR℄).Expliitlythequadratibraketf;g

Q

andtheubi

braketf;g

C

aregiven forF;G2O(g)by

fF;Gg

Q (x)=

1

2

h[x;rF(x)℄;R (xrG(x)+rG(x)x)i

g 1

2

h[x;rG(x)℄;R (xrF(x)+rF(x)x)i

g

fF;Gg

C

(x)=h[x;rF(x)℄;R (xrG(x)x)i

g

h [x;rG(x)℄;R (xrF(x)x)i

g :

When appliedto theR -braketon theloopalgebra

~

gofg=g l(N) we geta quadratianda ubi

PoissonbraketonO(

~

g ). Itwasshownin[LP℄thatthelinear,thequadratiand theubibraket

arerelated bythevetor eldU

X()

=X 2

(). Itis easy to prove thatU is amaster symmetryfor

thealgebraA, whihis inthease g=gl(N) generatedbythefuntions

I

ij

(X)=Res TrX

i

()

j+1

; i>0;j2Z;

[LP℄givesLenardrelations forthefuntionsI

ij

withrespetto thesebrakets. Usingthefatthat

U andV ommuteitiseasyto showthatV alsohasthedeformationpropertywithrespetto both

thequadratiand theubi brakets, e.g., theLiederivativeinthediretionof V ofthequadrati

l l 1

thisleads inpartiular to another set of Lenard relations for thefuntions I

ij

. It follows that on

the loop algebra

~

g the ubi and the quadrati braket have all properties whih the R -brakets

have: A is involutive with respet to these brakets, the orresponding Hamiltonian vetor elds

are multi-Hamiltonian with respet to these brakets and the brakets are onneted by the Lie

derivativewithrespettothevetor eldsU andV whiharemastersymmetriesforA. Thehigher

order brakets dier however from the linear strutures inone ruialaspet: asit is easy to see

they do not restrit to any of the nite-dimensional spaes

~

g

p;q

, dened in (2.4). Similarlythe

vetor eld U learly does not restrit to any of the subspaes

~

g

p;q

(exept in the trivial ase

p=q=0).

In order to onstrut our examples we need a redution theorem whih leads to a Poisson

strutureinthefollowingsituation: forN asubvarietyofan aÆnePoissonvariety(M;f;g) with

an algebrai group G ating on it (leaving N stable) we want an inherited Poisson struture on

the quotient spae N=G. By an aÆne Poisson varietywe mean an aÆne varietywhose algebra of

regular funtions is equipped with the struture of a Poisson algebra. We will assume that our

groupG also arriesaPoissonstruture(whih maybetrivial).

IfN isaPoissonsubvarietyofM thena Poissonstrutureon N=G,or,more preisely,onthe

ringO(N) G

ofG-invariantregularfuntionsonN,willexistifthemap:GN !N isaPoisson

map with respetto some Poisson struture on G; suh an ation is alled a Poisson ation 4

and

thebraketisalled a reduedbraket. IfN is nota Poisson subvarietyof M thenN=G may still

inheritabraketfromM: wewillgivebelowneessaryand suÆientonditionsforthistohappen.

The following notation will be useful: the algebra of regular funtions on M whih restrit

to G-invariant funtions on N is denoted by O(M;N) G

; we have a natural restrition 5

map :

O(M;N) G

!O(N) G

. TheidealofN isdenotedbyI(N)andwehaveaninlusionmap{:N !M.

Also, if:M

1

! M

2

is a regular mapbetween aÆne varietiesthen we denote by

the indued

mapO(M

2

)!O(M

1

) denedby

(f)=fÆ.

Denition 3.1 Let ( M;f;g) be an aÆne Poisson variety, :GM ! M a Poisson ation

and N asubvarietyof M whih is G-stable. Then the triple(M;G;N) isalled Poisson-reduible

ifO(M;N) G

isaPoissonsubalgebraofO(M)andifthereexistsaPoissonbraketonO(N) G

suh

that

f(F

1 );(F

2 )g

O(N)

G =fF

1

;F

2

g (3:1)

holdsforall F

1

; F

2

2O(M;N) G

.

Formula(3.1) says that inorder to omputethePoissonbraketof two G-invariant funtions

onN one omputes thePoissonbraketofanyextensionsto M andthenrestritstheresultto N.

Note alsothat (3.1) uniquelydenes a braket onO(N) G

(ifit exists) sine issurjetive. Inthe

following theorem, whih is similar in spiritto theMarsden-Ratiu redution theorem (see [MR℄),

we give neessaryand suÆient onditionsfor(M;G;N) to be Poisson-reduible.

Theorem 3.2 Let (M;f;g) be an aÆne Poisson variety, : GM ! M a Poisson ation

and N a subvarietyof M whih is G-stable. Then(M;G;N) is Poisson-reduible if and only if

O(M;N) G

;I(N) =0; (3:2)

it is impliit in this ondition that its left hand side makes sense.

Proof

Supposerst thatondition(3.2) is satised. We proeedto showthat

O(M;N) G

;O(M;N) G

O(M;N) G

:

4

Some authors, e.g., [LM℄ use this term in the more restrited sense in whih G is given the

trivialPoissonstruture;thenbeingaPoissonationmeansthatforanyg2Gtheinduedmap

g

:N !N is aPoissonmap.

5

This restrition map is onto, although the restrition map O(M) G

! O(N) G

is not onto in

general.

2;N

funtionf 2O(N) isonveniently expressedbythe formula

f =

2;N

f. Thuswe need to show

that

{

fF

1

;F

2 g=

2;N {

fF

1

;F

2

g (3:3)

foranyF

1

;F

2

2O(M;N) G

. Sine and

2;N

are therestritionsto GN of theorresponding

maps and

2;M

on GM and sinethese mapsare Poisson maps,(3.3) is equivalentto

(1

G {)

f

F

1

;

F

2 g

GM f

2;M F

1

;

2;M F

2 g

GM

=0; (3:4)

where 1

G

is the identity map on G. For g 2 G and n 2 N we dene maps

g

: M ! M and

n

:G!M byinsertingg resp.n in. Then

n

F isonstant foranyF 2O(M;N) G

sothat

f

F

1

;

F

2

2;M F

2 g

GM

(g;n)=f

g F

1

;

g F

2 F

2

g(n)+f

n F

1

;

n F

2 F

2 (n)g

G (g)

=f

g F

1

;

g F

2 F

2 g(n);

whih vanishesbytheassumption (3.2). Therefore

(1

G {)

f

F

1

;

F

2

2;M F

2 g

GM

=0; (3:5)

and similarly

(1

G {)

f

F

1

2;M F

1

;

2;M F

2 g

GM

=0: (3:6)

Summing(3.5) and (3.6) we nd(3.4) whihshowsthat fF

1

;F

2

g2O(M;N) G

.

Itfollows thatwe an atuallyuse(3.1) todene f;g

O(N)

G: ontheone handis surjetive,

on theotherhandthebraketonO(N) G

given by(3.1) is well-denedsineif(

~

F

2

)=(F

2 ) then

{

fF

1

;

~

F

2 F

2

g = 0; another appliation of (3.2). From the denition it is also immediate that

f;g

O(N)

G satisestheJaobiidentitysowegetaPoissonbraketonO(N) G

whihsatises(3.1).

Thisshows theifpart;theonlyifpart istrivialsine(I(N))=0.

Remark 3.3 Supposethatall algebras underonsiderationarenitelygenerated. Then O(N) G

is thealgebraof funtions on anaÆne varietyN=Gwhihan be onsideredasthe quotient of N

byG. SimilarlyO(M;N) G

orrespondsthentoanaÆnevariety(M;N)=G,obtainedbytakingthe

quotient of M with respet to G but along N only, i.e., onlyN is shrunk inside M into its orbit

spae N=G while the other points of M remain intat. In geometri terms formula (3.2) states

that the Hamiltonian vetor eldswhih areassoiated to funtions on M whih areG-invariant

onN,aretangenttoN (atpointsofN). Itfollowsfrom theproofofTheorem3.2thatifondition

(3.2) holdsthen(M;N)=G inheritsa PoissonbraketfromM and inturn N=G inheritsaPoisson

braketfrom(M;N)=G, thelatter beause all Hamiltonianvetor eldson (M;N)=G aretangent

to N=G.

As an appliation of this theorem let us show that if a vetor eld V whih desends to the

quotient has the deformation property with respet to some Poisson-reduible braket then this

deformationpropertyis onservedafter theredution. We needthefollowinglemma.

Lemma 3.4 Let M be an aÆne variety, V a vetor eld on M and G a linear algebrai group

ating on M; let N be an aÆne subvariety, stable for G, and suppose that V is tangent to N,

W =V

jN

. Then

L

W O(N)

G

O(N) G

(3:7)

L

V

O(M;N) G

O(M;N) G

(3:8)

and implies the ommutativity of the following diagram.

O(M;N) G

! O(N) G

L

V

?

?

y

?

?

y L

W

O(M;N) G

!

O(N) G

(3:9)

Proof

Let

N

: O(N) G

! O(N) denote the inlusion map (whih may be thought of as oming

from the quotient map

N

:N ! N=G) and note the obvious relation {

=

N

, whih holds on

O(M;N) G

. Then formula (3.8) followsfrom (3.7),

{

L

V

O(M;N) G

=L

W {

O(M;N) G

=L

W

N O(N)

G

=

N L

W O(N)

G

N O(N)

G

;

fortheproofof theother diretionsurjetivityof isessential:

N L

W O(N)

G

=L

W

N

O(M;N) G

={

L

V

O(M;N) G

{

O(M;N) G

=

N O(N)

G

:

Moreover, for F 2O(M;N) G

we have

N L

W

(F)=L

W

N

(F)=L

W {

F ={

L

V F =

N L

V F;

whih shows thatthediagram is ommutative.

Theorem 3.5 Let (M;G;N) be Poisson-reduible with respet to a Poisson braket f;g on M

and suppose that V is a vetor eld on M whih is tangent to N;W = V

jN

, and whih has the

deformation property withrespet tof;g. If L

W O(N)

G

O(N) G

then

a) (M;G;N) is Poisson-reduible with respet to f;g 0

, the Lie derivative of f;g in the

diretion of V;

b) W has the deformation property with respet to f;g

O(N) G;

) the Liederivative of f;g

O(N)

G in the diretion of W is the redued braket of f;g 0

.

Thus the deformationpropertysurvivesthe redution andthe operations ofredutionanddeforma-

tion ommute.

Proof

To show that (M;G;N) is Poisson-reduiblewith respetto f;g 0

, we usethe neessary and

suÆient ondition(3.2) ofTheorem3.2. Sine

fF;Gg 0

=L

V

fF;Gg fL

V

F;Gg fF;L

V Gg;

we have that

fO(M;N) G

;I(N)g 0

=L

V

fO(M;N) G

;I(N)g fL

V

O(M;N) G

;I(N)g fO(M;N) G

;L

V I(N)g

and eahtermof theright handsidevanishesbeause (M;G;N) isPoisson-reduiblewithrespet

to f;g: fortherst termuse ommutativity of (3.9),for theseond one use(3.8) and thelastis

zero beause V istangent to N;L

V

I(N)=0.

O(N)

f;g 0

O(N) G

off;g 0

. This meansthatiff

1

;f

2

2O(N) G

then

ff

1

;f

2 g

0

O(N) G

=L

W ff

1

;f

2 g

O(N) G fL

W f

1

;f

2 g

O(N) G ff

1

;L

W f

2 g

O(N)

G: (3:10)

Letf

1

=(F

1 ); f

2

=(F

2

) and use(3.1) and ommutativityof (3.9):

ff

1

;f

2 g

0

O(N) G

=fF

1

;F

2 g

0

=L

V fF

1

;F

2

g fL

V F

1

;F

2

g fF

1

;L

V F

2 g

=L

W ff

1

;f

2 g

O(N) G fL

W f

1

;f

2 g

O(N) G ff

1

;L

W f

2 g

O(N) G:

Sinewehave proved thatf;g 0

O(N) G

isa Poissonbraket on O(N) G

we have showninpartiular

thatL

W

hasthedeformation propertywithrespetto f;g

O(N)

G and we aredone.

Remark 3.6 Under theonditions ofRemark 3.3 theonditions (3.7) and (3.8) mean thatthe

vetor eldsW and V aretangent to the quotient spaesN=G and(M;N)=G.

Remark 3.7 TheonditionsofTheorem3.5arealsosuÆientto onludethatamastersymme-

tryforasubalgebraAO(M;N) G

desendsto amastersymmetryonthequotient. Toprove this

let F 2O(M;N) G

andnote thatX

F

=f;Fg istangent to N. If we denote byY

F

therestrition

of X

F

to N thenY

F

isgiven as aderivationof O(N) G

byY

F

=f;(F)g

O(N)

G and we have that

Y

F

=X

F

. Using(3.9), writtenasW =V,we get

[Y

F

;[ Y

G

;W℄ ℄=[X

F

;[ X

G

;V℄ ℄=0;

sineV isa master symmetryforA. Sine issurjetive W isa mastersymmetry for(A).

Inthissetion weapplyourtworedutiontheoremsto thenite-dimensionalsubspaes

~

g

n of

~

g,denedin(2.5), inase

~

gistheloopalgebraof aomplexsimpleLiealgebraggl(N)of rank

r (see[Hum℄ and[Ser℄). We denotebyGanyalgebraigroupwhoseLiealgebraequals g. Wexa

WeylbasisfH

i

;E

i

;F

i g

r

i=1

ofg,i.e.,aolletionof3r generatorsforg suhthatH

i

spansaCartan

subalgebrah,and thefollowingommutationrelations 6

hold:

[E

i

;F

j

℄=Æ

ij F

i

; [H

i

;E

j

℄=n

ji E

j

; [H

i

;F

j

℄= n

ji F

j :

Here (n

ij

) is the Cartan matrix of g and the indiesi;j take values between 1 and r. The Weyl

basis leads to a gradation g = k

i= k g

i

of g: g

0

= h and for i positive (negative) g

i

is spanned

by the i-fold ommutators of the elements E

1

;:::;E

r (F

1

;:::;F

r

). An element of g

i

is alled a

homogeneouselementofdegreeiandh=k+1isalledtheCoxeternumberofg. Theprojetionof

gong

i

isdenotedby

i

. Wewillalsousethedeompositiong=n hn +

,wheren

= k

i=1 g

i .

WewillonsiderasetfI

1

;:::;I

r

gofChevalleyinvariantsofg . Theyarehomogeneouspolynomials

whih generatethealgebra O(g ) G

of invariantsfortheadjoint ationofGon g (see[Var℄p. 333).

We denote the degree of I

j by d

j

and all the numbers q

j

= d

j

1 the exponents of g . We will

always assume the invariants I

j

to be ordered bydegree. Then the exponents bearthe following

relations(see [Kos1℄):

1=q

1

<q

2 q

3

q

r 1

<q

r

=k: (4:1)

The Chevalleyinvariantslead to thefollowingG-invariantfuntions on

~

g :

I

ij

(X)=Res I

i (X)

j+1

;

whih bydenitiongenerate theinvolutive algebraA introduedinSetion2.

4.1. Poisson redution

Let and be homogeneouselements of gsuhthat deg deg =h. We putdeg = d

and we dene, asinParagraph 2.1,

~

g

n

= (

n

X

i=0 x

i

i

2

~

gjx

n

= )

;

together withthefollowingaÆne subspae

N = (

n

X

i=0 x

i

i

2

~

g

n j

j (x

n 1

)=0 ifjdeg )

:

Lemma 4.1 Let g

be the isotropy algebra of and let g

= g

\n . Then the ation of

G

=expg

on

~

g

n

leaves N invariant.

6

Our denition of a Weyl basis diers from the one in [Ser℄ by a transpositionin the Cartan

matrix, i.e., [Ser℄ takes [H

i

;E

j

℄ = n

ij E

j

; our hoie simplies the expliitformulas for the Weyl

bases given intheexamples.

ItsuÆesto showthat if

j (x

n 1

)vanishesforall jdegthenthesame holdstruefor

j (Ad

exp x

n 1

)when 2n . The result followsat one fromAd

exp

=expad

:

ReallfromProposition2.3thatthebraketsf;g

l

restritto

~

g

n

for0ln. Notiehowever

that the braket f;g

n

does not restrit to N: if e

a

is a basis element suh that dege

a

= deg

then n 1

a

a

() belongsto the ideal I(N) of N but f n 1

a

; 0

b g

n

=C

ab

0

, whih is non-zerofor

at least one value of b sine g is simple. Therefore we are preisely in the ase of the redution

theorem (Theorem3.2).

Theorem4.2 Thetriple

~

g

n

;G

;N

isPoisson-reduiblewithrespettoeahPoissonstruture

P

n

l=0

l f;g

l .

Proof

We rst show thatthe ation of G

on

~

g

n

is Poisson. To do thiswe take thetrivialPoisson

struture on G, we x any l 2Z and show that (Ad

g )

ff

1

;f

2 g

l

= f(Ad

g )

f

1

;(Ad

g )

f

2 g

l

for any

g2G and anyf

1

;f

2 2O(

~

g ). It issuÆient to show thisforf

1 and f

2

linear; then(Ad

g )

f

1 and

(Ad

g )

f

2

arelineartoo andtheirgradientsdonotdependonX 2

~

g(inpartiularwean omitthe

argument). Sine

d

dt

jt=0 f

1 (Ad

g

(X+tY))=f

1 (Ad

g

Y)=h rf

1

;Ad

g Yi

we ndthathr(Ad

g )

f

1

;Yi=

Ad

g 1rf

1

;Y

givingr(Ad

g )

f

1

=Ad

g 1rf

1 . Then

f(Ad

g )

f

1

;(Ad

g )

f

2 g

l (X)=

X;[Ad

g 1rf

1

;Ad

g 1rf

2

℄

Rl

=hAd

g

X;[rf

1

;rf

2

℄

Rl i

=ff

1

;f

2 g

l (Ad

g X)

=(Ad

g )

ff

1

;f

2 g

l (X):

We will see in Proposition 4.4 that in ertain interesting ases the ation is even Hamiltonian.

We now verify ondition (3.2). The ideal I(N) of N is generated by those elements of the form

n 1

a

a

() forwhihdege

a

d h. For l=0;:::;n 1 these elements areCasimirs of f;g

l .

Indeed, ifX2

~

g

n

and b2I and 0k n 1 then

f n 1

a

; k

b g

l

(X)= n 1;k

l C

ab

n+k l

(X)=0

fork 6=l,whileifk=lthen

f n 1

a

; l

b g

l

(X)=C

ab

n

(X)=C

ab he

;i

g

=h[e

a

;e

b

℄;i

g

=h[;e

a

℄;e

b i

g

=0:

We used inthelastequalitythat [;e

a

℄=0ifdege

a

d h,whihfollows from deg[;e

a

℄ h.

Thisshows that(3.2) is satisedwhen l6=n. Asforthen-thbraket, letF 2O(

~

g

n

;N) G

and let

a be suh that dege

a

d h; notiethat if F restritsto a G

invariant funtionon N then F

satisestheinnitesimalondition

h[rF(X);X℄;i=0; 8X2N;8 2g

:

We needto showthatf

a

;Fg

n

(X)=0foranyX 2N. But

f n 1

a

;Fg

n (X)=

1

2

X;[e

a

;rF(X)℄+[ n

e

a

;R ( n

rF(X))℄

= 1

2

X;[e

a

;rF(X)℄+[ n

e

a

; n

rF(X) 2(

n

rF(X)) ℄

=hX;[e

a

;rF(X)℄i

X;[ n

e

a

;( n

rF(X)) ℄

:

For therst term we have h X;[e

a

;rF(X)℄i=h e

a

;[rF(X);X℄i=0 sine e

a 2g

, again beause

[;e

a

℄=0. Similarlywendthat theseond termvanishes:

X;[ n

e

a

;( n

rF(X)) ℄

=h;[e

a

;( n

rF(X))

1

℄i

g

=0:

If is a generi nilpotent element then the ation of G

is Hamiltonian, a fat that an be

used to give an alternative proof ofTheorem 4.2for suh . The proofof thisdepends on several

fats about simple Lie algebras whih we will reall now (see [Kos1℄ for details). A nilpotent

element is alled prinipal when its isotropy algebra has dimension r; in this ase the isotropy

algebra is Abelian. A generi nilpotent element is prinipal and all prinipal nilpotent elements

are onjugate to = P

r

i=1 F

i

whose isotropy algebra g

is ontained in n . Notie that as a

onsequeneG

=G

andthusthatitsuÆesto provethattheationofG

on

~

g

n

isHamiltonian

for= P

r

i=1 F

i

. Wewillusethefollowinglemmaaboutthegradientsof theChevalley invariants.

Lemma4.3 If = P

r

i=1 F

i

then the gradientof the i-th Chevalleyinvariant I

i

at is homoge-

neous of degree (d

i

1) and the gradients rI

1

();:::;rI

r

() are linearly independent.

Proof

FortherstlaimitsuÆestoshowthathrI

i ();yi

g

=0forally2g

j

withj 6=(d

i

1),sine

hg

l

;g

m i

g

=0 ifl+m6=0. To showthisweintrodueforall x2g theoperator

x

:O(g)!O(g )

denedby

(

x

f)(z)=hrf(z);xi= d

dt

jt=0

f(z+tx);

andweobservethatg(x)= 1

m!

( m

x

g)(0) foranyhomogeneouspolynomialg ofdegreem. Then,for

x= andf =I

i

,we get

hrI

i

();yi=

y I

i ()=

1

(d

i 1)!

( di 1

y I

i )(0):

But the proof of Lemma 14 in [Kos2℄ shows that ( d

i 1

y I

i

)(0) = 0 if y 2 g

j

with j 6= d

i 1.

Finally,theelementsrI

i

() arelinearlyindependentsinedimg

=r (see [Kos2℄, Theorem9).

Proposition4.4 If = P

r

i=1 F

r

thenthe adjointationof G

on

~

g

n

isHamiltonianwithrespet

to every Poisson struturef;g

l

;l=0;:::;n. We an hoose a basisfb

i g

r

i=1 of g

in suh a way

that the orresponding innitesimal generators X

i

of the ation arethe Hamiltonian vetor elds

X

i

=f;I

i;n(d

i 1) 1

g

0

=f;I

i;n(d

i

1)+l 1 g

l

; l=0;:::;n; i=1;:::;r:

We rst showthat forany X2

~

g

n

and i=1;:::;r

b

i

:= rI

i;n(d

i 1) 1

(X)

+

(4:2)

isindependentof X aswellasof and belongsto g

. Tosee this,take x2g to ndthat

rI

i;n(d

i 1) 1

(X);x l

= d

dt

jt=0 Res

I

i

(X+tx l

)

n(d

i 1)

=Res 1

n(d

i 1)

d

dt

jt=0 I

i (

n

+:::+x

0 +tx

l

);

=Res n

d

dt

jt=0 I

i (+x

n 1

1

+:::+x

0

n

+tx l n

)

=Res l

hrI

i (+x

n 1

1

+:::+x

0

n

);xi

g :

Developing rI

i

in a Taylor series at we nd (taking l 2) that b

i

is independent of , and

(taking l=1) thatb

i

=rI

i

(), independent ofX. From thisdesriptionwe mayonludeon the

one handthattheelementsb

i

areindependent,asaorollary ofLemma4.3; ontheother handwe

mayonlude thateah b

i

belongsto theisotropyalgebrag

of ,sine

[;b

i

℄=[;rI

i

()℄=0

byAd-invarianeofI

i

. Sine dimg

=r,it followsthat theb

i

(i=1;:::;r) spang

.

The orresponding generators are learly the vetor elds X

i

dened by _

X = [b

i

;X℄, where

X2

~

g

n

. Butusing(2.8) and thedenitionofb

i

itis easilyseen thatX

i

istheHamiltonianvetor

eldassoiatedwithI

i;n(d

i 1) 1

bymeansoff;g

0

. Moreover, sinetheirHamiltoniansareofthe

form I

ij

the ation of G

is atually Hamiltonian with respet to any of the Poisson strutures

f;g

l

(l=0;:::;n).

4.2. Redution of the master symmetry

WenowturnourattentiontothevetoreldV on

~

gwhihwasshowntobeamastersymmetry

forAand to have thedeformationpropertywithrespettothebraketsf;g

l

. Wewillnowshow

thatV desends to amaster symmetry whihhasthe deformationproperty. We willusethe same

notation f;g

l

forthe reduedbrakets (onO(N) G

) asfortheoriginalones(on O(~

g

n )).

Proposition 4.5 Themaster symmetry V istangent to

~

g

n and L

W O(N)

G

O(N) G

, where

W denotes the restrition of V to N. Therefore, the brakets f;g

l

and f;g

l 1

on O(

~

g

n

) whih

are onneted by the Lie derivative with respet to V redueto two brakets f;g

l

and f;g

l 1 on

O(N) G

whih are onneted by the Lie derivative with respet to W. Moreover W is a master

symmetry for (A).

Proof

Theowof V isgiven by

s

:X()7!X(+s);

ifX()2N;X()= +x

n 1

+ , then

X(+s)=(+s) n

+x

n 1

(+s) n 1

+

=

n

+(ns+x

n 1 )

n 1

+

(4:3)

belongsto N sine 2 n , showing that V is tangent to N. Also it is lear that L

W O(N)

G

O(N) G

beause G

ats by simultaneous onjugation on the oeÆients of in X(), hene

ommutes with L

W

. ThusTheorem3.5 appliesto yield the rst statement. The fatthat W is a

master symmetryfor(A) follows fromRemark 3.7.

4.3. The redued spae N=G

as a linear subspae N

0 N

We now show that when + is regular then the algebra O(N) G

is nitely generated

(although G

is notredutive) and that the quotient spae N=G

an be identied in a natural

way with an aÆne subspae N

0

of N. By naturality we mean here that under the identiation

whih we willonstrut the involutive algebra A O(N=G

)

= O(N)

G

and the vetor eld W

orrespondto theirrestritiontoN

0

,heneaneasilybeomputed. NotehoweverthatthePoisson

strutures f;g

l onN

0

are notobtainedby restrition.

We willassume, asinParagraph4.1, that and are homogeneouswith deg deg =h.

Weputd= deg andweassumethat =+ isregular,meaningthattheisotropysubalgebra

of is aCartan subalgebra. We will give at theendof thissetion forevery simpleLiealgebra g

an important lass of pairs (;) suh that + is regular. The onlyproperty that we willuse

abouttheregularityof+ is ontainedinthefollowinglemma.

Lemma 4.6 Let and beas above. Then g

\g

=f0g.

Proof

Ifx2g

\g

,thenxbelongsto theisotropyalgebraof +,whihisa Cartansubalgebra,

hene xis semisimple. Ontheother handx2n hene itis nilpotent. Thereforex=0.

ThespaeN

0

isonstrutedasfollows. Letq

i

beasubspaeofg

i

,fori=1 d;:::;h d 1,

suh that

g

i

=q

i

g

\g

i+d h

;

: (4:4)

Ifwe denote

q= d

i= k g

i

h d 1

i=1 d q

i

;

thenN

0

is denedby

N

0

=fX 2N jx

n 1

=+x~

n 1

;x~

n 1 2qg:

Theorem4.7 If + isregularthen the inlusion |:N

0

!N indues analgebra isomorphism

O(N) G

= O(N

0

)sothat N=G

isanaÆnespaewhihanbeidentiedwithN

0

:Thefuntionsin

involutionI

ij

andthemastersymmetryW on N

0

aretherestritionsof theorrespondingfuntions

I

ij

andthe master symmetry W on N.

We rst dene a regularmap N !G

:X 7!g

X

whih has thepropertythat Ad

g

X

X 2N

0

foranyX 2N and equals X foranyX 2N

0

. To determineg

X

usethe fatthat g

isontained

inn to writeitas

g

X

=exp ; with = k

X

j=1

j 2g

;

j 2g

\g

j :

Then

(Ad

gX )

n 1

=Ad

gX x

n 1

=x

n 1 +[ ;x

n 1

℄+

= + k d

X

i= k

i x

n 1

!

+ 2

4 k

X

j=1

j

;+ k d

X

i= k

i x

n 1 3

5

+;

whihhastobeequalto + P

h 1 d

i=1 d q

i

+p;withq

i 2q

i

andp2 d

i= k g

i

. Theprojetionong

h d

yields= whileprojetionon g

h d 1

leadsto

h d 1 x

n 1

=q

h d 1 [

1

;℄;

from whihq

h d 1 and

1

are uniquelydetermined beause of thediretsum deomposition

g

h d 1

=q

h d 1

g

\g

1

;

and g

\g

=f0g (Lemma 4.6). More generally,the projetionon g

j

(j =h d 1;:::;1 d)

yields

j x

n 1

+(known stu)=q

j [

j+d h

;℄;

whih gives a unique q

j 2q

j

and a unique

j+d h 2g

\g

j+d h

. This gives usthe desired map

N !G

;sineallq

j and

i

areunique,allelementsofN

0

maptotheidentityelementinG

. The

mapN !G

isregular beause the

i

dependlinearlyon theentriesof x

n 1

and exp:g

!G

is a regularmap. Notie thatonly

j x

n 1

;j=1 d;:::;h d 1 enter theonstrution of g

X

;

sothat g

X

=g

X 0

if

j x

n 1

=

j x

0

n 1

forj =1 d;:::;h d 1.

We thus also have a regular map :N ! N

0

given by X 7! Ad

gX

X. Let usshow that the

image ofthe induedinjetive map

:O(N

0

)!O(N) is preiselyO(N) G

. IfF 2O(N

0 ) then

F is G

-invariant beause is G

-invariant, hene

is injetive; also, ifF isa G

-invariant

funtion then its restrition to N

0

maps to F under

, hene the image of

is O(N) G

. In

onlusionO(N

0

)and O(N) G

are isomorphiandwe an identifyN

0

asthequotient N=G

.

ThefuntionsI

ij

onN areG

-invarianthenepasstothequotientN

0

. Sinethequotientmap

wasinduedbytheinlusionmap|:N

0

!N theorrespondingfuntions onN

0

arejustobtained

byrestrition. Formula(4.3) impliesthatW istangentto N

0

and alsothatg

X()

=g

X(+s) . Ifwe

denote therestrition of W to N

0 byW

0

then itfollows that L

W

=

L

W 0,

inother wordsthe

projetionofW onN

0

=N=G

isjustW 0

,therestritionofW toN

0

. Inonlusionthefuntions

ininvolutionand theirmaster symmetryhave asimpledesriptionon thereduedspae N

0 .

Weendthissetionbygivingageneralruletoseletpairs(;),withdeg deg=h,suh

that+isregular. Werstreallsomefatsfrom[Kos1℄,Theorem6.7. Let

1

= P

r

i=1 F

i

andlet

1 2g

k ,

1

6=0. Then

1

=

1 +

1

isregular,sothath =g

1

isa Cartansubalgebra. Moreover,

there existsabasisf

1

;:::;

r g ofh

0

with thefollowingproperties(here

istheprojetiononto

n

and q

1 q

2

:::q

r

are theexponentsof g):

1)

s

= (

s

) ishomogeneousof degree q

s

;

2)

s

= +

(

s

)is homogeneousof degree h q

s .

Thenextpropositionallows usto determinewhihpairs(

s

;

s

) aresuhthat

s

=

s +

s is

regular. We aregrateful to B.Kostant forproviding uswitha proof.

Proposition 4.8 Theelement

s

isregularifand onlyif q

s

isoprime tothe Coxeternumber h.

Proof

Let H

0

be the unique element inh suh that [H

0

;E

i

℄= E

i

for all i=1;:::;r (the existene

and uniqueness of H

0

follow from the fat that the Cartan matrix of g is invertible). Then it is

easilyseen that

[H

0

;x℄=jx 8x2g

j

: (4:5)

If we dene P

0

2 G by P

0

= exp(

2 p

1

h H

0

), then (4.5) implies that Ad

P0

i

= ! q

i

i

, where

! = e 2

p

1 =h

. In partiular we have that Ad

P0 (h

0

) h 0

, so that P

0

belongs to the normalizer

N(H 0

) of H 0

=exph 0

in G. We denote the element of N(H 0

)=H 0

whih orresponds to P

0 by.

The groupW =N(H 0

)=H 0

is alledthe Weylgroup ofg . Clearlyeah element w2W ats on h 0

bytheadjoint ation; we willusew(x) to stand forAd

g

x, whereg2N(H 0

) isanyrepresentative

of w. Sine q

1

=1,it followsfrom thefat that(

s )=!

q

s

s

fors=1;:::;r thattheorder of

ish.

Nowletussupposethat m>1is aommon divisorofq

s

and h. Thenwean writeh=h 0

m,

q

s

=q 0

s

mforsome h 0

;q 0

s

2N. Weshowthat

s

annotberegularbyprovingthat h

0

isanontrivial

elementof W that leaves

s

xed(see,e.g., [Kna℄,p.426{427). Indeed,

h

0

(

s )=!

q

s h

0

s

=! hq

0

s

s

=

s

;

and h

0

is notidentitybeause theorder of ish>h 0

.

Conversely,assumethatq

s

isoprimetoh. Then!

s

=! q

s

isstillaprimitiveh-root ofunity.

If fI

j g

j=1;:::;r

are the Chevalley invariants, degI

j

= q

j

+1, then we have that I

j (

s

) = 0 for all

j<r. Indeed,

I

j (

s )=I

j ((

s ))=I

j (!

s

s )=!

q j

+1

s I

j (

s );

while(4.1) impliesthatq

j

+1<q

r

+1=h,sothat! q

j +1

s

6=1forj<r. Ontheotherhand,I

r (

s )

annot vanish, beause any element of g at whih all invariants vanishis nilpotent (see Theorem

9.1of [Kos1℄). Thereforethere existsa non-zero b2Csuhthat I

j (b

1 ) =I

j (

s

) forj=1;:::;r.

Now, Lemma9.2of [Kos1℄ statesthattwoelementsofa Cartansubalgebraat whih allinvariants

take the same valuesare W-onjugate. Sine

1

is regular,

s

isregular too.