A NNALES DE L ’I. H. P., SECTION A
Y VETTE K OSMANN -S CHWARZBACH F RANCO M AGRI
Poisson-Nijenhuis structures
Annales de l’I. H. P., section A, tome 53, n
o1 (1990), p. 35-81
<http://www.numdam.org/item?id=AIHPA_1990__53_1_35_0>
© Gauthier-Villars, 1990, tous droits réservés.
L’accès aux archives de la revue « Annales de l’I. H. P., section A » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.
org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
http://www.numdam.org/
35
Poisson-Nijenhuis structures
Yvette KOSMANN-SCHWARZBACH
Franco MAGRI
U.F.R. de Mathématiques, U.A. au C.N.R.S. 751, Universite de Lille-1, 59655 Villeneuve d’Ascq, France
Dipartimento di Matematica
Universita di Milano, 1-20133 Milano, Italy
Vol. 53, n° 1,1990, Physique théorique
ABSTRACT. - We
study
thedeformation,
definedby
aNijenhuis
opera- tor, and thedualization,
definedby
a Poissonbivector,
of the Lie bracket of vector fields on a manifoldand,
moregenerally,
of the Lie bracket on a differential Liealgebra
over a commutativealgebra. Requiring
that thetwo processes commute, one obtains hierarchies of
pairwise compatible
Lie brackets on the module and on its dual. Each differential Lie
algebra-
structure on a module
gives
rise to acohomology
operator on thealgebra
of forms over the
module,
as well as to agraded
Liealgebra-structure
onthe
algebra
of multivectors(the
Schoutenalgebra).
Westudy
the deforma- tion and the dualization of the derivations of thealgebra
of forms and ofthe Schouten bracket of
multivectors,
thusobtaining generalizations
ofthe
preceding
differentialgeometric
resultstogether
with newproofs.
Section 2
comprises
thestudy
of theNijenhuis
operators on the twilled Liealgebras,
an "N-matrix version" of theKostant-Symes theorem,
andan
application
to Hamiltonian systems of Toda type onsemisimple
Liealgebras.
RESUME. 2014 Nous etudions la
deformation,
definie par un tenseur deNijenhuis,
et ladualisation,
definie par un bivecteur dePoisson,
du crochetde Lie des
champs
de vecteurs sur une variete et,plus generalement,
du crochet de Lie sur une
algebre
de Lie differentielle sur unealgèbre
commutative. En
imposant
la commutativité des deuxoperations,
onAnnales de l’lnstitut Henri Poincaré - Physique théorique - 0246-0211 1
’
obtient,
sur le module et sur sondual,
des hierarchies de crochets de Lie deux à deuxcompatibles.
Toute structured’algèbre
de Lie différentiellesur un module
correspond
d’une part à unopérateur
decohomologie
surl’algèbre
des formes sur le module et, d’autre part, à une structured’algèbre
de Lie
graduée
surl’algèbre
des multivecteurs sur le module(algebre
deSchouten).
Nous étudions la deformation et la dualisation des derivations del’algèbre
des formes et du crochet de Schouten desmultivecteurs,
obtenant ainsi des
généralisations
de résultatsprécédents
degéométrie différentielle,
ainsi que des démonstrations nouvelles. Auparagraphe 2,
nous étudions les
opérateurs
deNijenhuis
sur lesalgèbres
de Liebicroisées,
une « version N-matrice » du théorème de
Kostant-Symes,
et uneapplica-
tion à des
systèmes
hamiltoniens du type de Toda sur lesalgèbres
de Liesemi-simples.
Table of Contents
Introduction
1 . Deformation of Lie brackets by means of a Nijenhuis operator 1.1. Constructing deformed brackets from Nijenhuis operators.
1. 2. Richardson-Nijenhuis bracket of E-valued forms on a vector space E and the properties of the deformed bracket.
1. 3. Properties of the iterated brackets.
1.4. Abelian subalgebras.
2. Deformation of Lie brackets and Hamiltonian systems 2. 1. The Nijenhuis operator of a twilled Lie algebra.
2. 2. The N-matrix approach to the Kostant-Symes theorem.
2. 3. Nijenhuis operators on semisimple Lie algebras and the Toda lattice.
3. Dualization of Lie brackets by means of a Poisson bivector 3. l. Poisson manifolds.
3. 2. The Lie bracket of I-forms on a Poisson manifold.
3 . 3. Example: the Lie bracket of I-forms on the dual of a Lie algebra.
4. Deformation and dualization: the compatibility condition
4. l. Deformed Lie bracket of vector fields and associated bracket on forms.
4. 2. The compatibility condition: Poisson-Nijenhuis structures.
5. Hierarchies of brackets of a Poisson-Nijenhuis structure
5 . 1. The iterated Lie brackets on vector fields and on 1 forms.
5 .2. Deformation and dualization of iterated Lie brackets.
5. 3. The hierarchy of Poisson structures on a Poisson-Nijenhuis manifold.
6. Differential Lie algebras, graded differential algebras and Schouten algebras
6 . 1. Differential Lie algebras.
6. 2. The graded differential algebra of a differential Lie algebra.
Annales dP l’Institut Henri Poincaré - Physique theorique
6.3. The Schouten algebra of a differential Lie algebra.
6.4. Graded differential algebra and Schouten algebra associated with a deformed Lie
bracket.
6.5. Dualization of the Lie bracket by means of a Poisson bivector on a differential Lie algebra. The associated Schouten algebra and graded differential algebra.
6.6. Morphisms of differential Lie algebras.
6.7. Conclusion: Poisson-Nijenhuis structures on differential Lie algebras.
INTRODUCTION
A
Poisson-Nijenhuis manifold ([30], [31])
is a manifoldequipped
withboth a Poisson structure, defined
by
a bivector P whose Schouten bracketvanishes,
and a(I,I)-tensor N
whoseNijenhuis
torsionvanishes,
to be called aNijenhuis
tensor, whichsatisfy
acompatibility
condition to bediscussed below. The
Poisson-Nijenhuis
structures on manifoldsand,
inparticular,
on duals of Liealgebras
constitute a natural framework for thetheory
ofcompletely integrable
systems. In work in progress, we shallstudy
therelationships
between thetheory
ofPoisson-Nijenhuis
structuresand that of Poisson-Lie groups,
bigebras
and the modifiedYang-Baxter equation
asexpounded
in[21].
This article can be considered as Part 0 of the series of papers[21]. Here,
we shalldevelop
thetheory
of Poisson-Nijenhuis
structures in aslightly
moregeneral
sisuation than that of smooth manifoldssince,
in section1,
westudy
theNijenhuis
structureson real or
complex
Liealgebras,
and weshow,
in section6,
that the restof the
theory developed
in sections3,
4 and 5 on smooth manifolds canbe extended to the case of the modules over a communitative
algebra
which are "differential Lie
algebras". (See
subsection 6.1 for definitions and historicalremarks.)
In section
1,
we consider an infinitesimal deformation of a Liealgebra-
strucure on a vector space
E,
and we provethat,
when such a deformation is definedby
anendomorphism
N of E whoseNijenhuis
torsionvanishes,
all the brackets obtainedby
iteration are Lie brackets which arepairwise compatible.
Section 2 studies therelationship
between theNijenhuis
opera- tors and the twilled Liealgebra-structures,
involution theorems on the dual of a Liealgebra and,
as anapplication,
some Hamiltonian systems of Toda type onsemisimple
Liealgebras.
In section
3,
we show how to construct a bracket on the space of 1-forms from the usual Lie bracket of vector fields and a bivectorP,
andwe prove that this bracket is a Lie bracket if and
only
if the Schoutenbracket, [P, P],
vanishes. The Lie brackket of 1-forms on a Poisson manifold was introducedby Magri
in[30], [31] ]
andindependently by
several authors.
(See
subsection3.2.)
For a
(I , I)-tensor N,
we have used thegenerally accepted
definition of theNijenhuis
torsion, formula(1 .1),
and we have denoted itby [N,
(We
remark that the bracketoriginally
definedby
Frolicher andNijenhuis
is twice this
quantity.)
For a bivectorP,
we have considered the Schouten bracket[P, P]~
definedby (3 . 1).
We haveincorporated
the factor 2 in the definition in order for this Schouten bracket to coincide with the oneconsidered in subsection 6 . 3. In both cases, the
index ~
refers to the Liealgebra-structure
on the space of vector fields.Apart
from the trivialquestion
ofnormalization,
the twoobjects play
roles which arestrikingly
similar: both measure the failure of a
mapping
to be amorphism
ofbrackets,
as shownby
formulae(1.6)
and(3.5).
When[N, N]~ vanishes,
N. Jl
is a Lie bracket and N is amorphism
from(E,
N ..Il)
to(E,
When[P, P]~ vanishes,
v(,, P)
definedby
formula(3.2)
is a Lie bracket on E*and P defines a
morphism
from(E*,
vP))
to(E, j~).
In section
4,
we combine the process of deformation of the Liealgebra-
structure of the linear space E of vector fields
by
means of aNijenhuis
tensor
N,
and that of the dualization of the Lie bracket on Eby
meansof a Poisson bivector P.
Requiring
that the two processes commute intro- duces thecompatibility
conditionlinking
N and P. The deformation pro-cess can be
iterated, and,
in section5,
we showthat,
when thecompatibility
condition is
satisfied,
a sequence of Lie brackets on the linear space E*of 1-forms is obtained in this way. We
study
thehierarchy
of Lie bracketson E and on E* obtained from a
Poisson-Nijenhuis
structure onE,
andwe prove that the iterated deformations commute with the dualizations and that both the Lie brackets on E and the Lie brackets on E* are
pairwise compatible.
In section
6,
we define andstudy
the differential Liealgebras
over acommutative
algebra A,
of which the C~M-module of vector fields on a manifoldM,
with the usual Liebracket,
is thetypical example.
Insubsections 6.1 and
6. 2,
we show how each differential Liealgebra
struc-ture on an A-module E defines a
graded
differentialalgebra-structure
onA
(E*), generalizing
the de Rhamcohomology
operator on the exterioralgebra
offorms,
and agraded
Liealgebra-structure
on AE, generalizing
the Schouten bracket on the exterior
algebra
of multivectors.In subsection
6. 4,
we showthat, given
an A-linearmapping
N from Eto
E,
the derivation of A(E*)
associated with the deformed bracketon E is the Lie derivation
(in
thegeneralized sense)
with respect to the E-valued 1-form N on E. We use this fact togive
analternate,
lesscomputational proof
of a theorem of section 1: If theNijenhuis
torsionof N
vanishes,
thenN. Jl
is a Lie bracket. We alsostudy
the deformedSchouten bracket on A E.
In subsection
6.5,
we show that the dualization of the Lie bracket of vector fields described in section 3 can be carried out moregenerally
on aAnnales de l’lnstitut Henri Poincaré - Physique théorique
differential Lie
algebra
E over a commutativealgebra
A. Given an A-linear, antisymmetric mapping, P,
fromE*,
the A-dual ofE,
toE,
thereis an associated bracket on E* which is a Lie bracket if and
only
if theSchouten bracket
of P,
considered as abivector,
vanishes. We prove(proposition 6.2)
that the derivation on multivectors associated with anA-linear, antisymmetric mapping
P from E* to E is theoperator [P, .],
where
[ , ]
denotes the Schouten bracket and P is the bivector definedby
P. When P is a Poisson bivector on amanifold,
this derivationis,
up to asign,
theG-cohomology
operator introducedby
Lichnerowicz[27].
We use this
proposition
to obtain ageneralization together
with analternate
proof
of the theorem of section 3: On anondegenerate
differential Liealgebra,
the bracket on E* associated with P is a Lie bracket if andonly
if the Schouten bracket of P vanishes. The Schouten bracket onA
(E*),
which extends the Lie bracket on E* definedby
the Poisson bivectorP,
is the bracket introducedby
Koszul in[23].
Since
morphisms
of differential Liealgebras correspond
tomorphisms
of
graded
differentialalgebras
between thealgebras
offorms,
we obtain aproof
of the fact that the exterior powers of the Poissonmapping
inter-twine the de Rham and the Lichnerowicz
cohomology
operators. More-over,
morphisms
of differential Liealgebras correspond
tomorphisms
ofSchouten
algebras and,
inparticular,
when P is a Poissonbivector,
AP isa
morphism
of Schoutenalgebras.
In
conclusion,
we stress the fact that thetheory
ofPoisson-Nijenhuis
structures
applies
to the case of the differential Liealgebras,
and weoutline some
applications
of thistheory.
Many
resultspresented
here werepreviously
formulated in various contexts. Forinstance,
the work of de Barros([ 10], [ 11 ])
is acomprehensive algebraization
of the infinitesimal structures on a manifold under moregeneral assumptions
than ours andincludes,
as anapplication,
the consi-deration of what we have called the deformed bracket
N. jn
and theproof
that the
vanishing
of theNijenhuis
torsion of Nimplies
the Jacobiidentity
for the deformed bracket
N. ~.
This fact was alsoproved
in[25].
It appears in a short announcementby
Dorfman[12]
withapplications
to thetheory
of
integrable
systems. We realized whilewriting
this article that the connection between the Lie bracket of differential 1-forms and the Lich- nerowiczcohomology
operator on a Poisson manifold had beenpreviously
noted in
[3], [4],
and apreprint by
Huebschmann[ 18]
came to our attentionin which a very
general algebraic
version of thetheory (but
not the factsabout the existence of the Schouten Lie
algebra
of a differential Liealgebra)
isdeveloped. Many previously
unrelated results from variousareas are
proved
in this article and used as thebuilding
blocks of thetheory
ofPoisson-Nijenhuis
structures.The main results of the
theory
had beenproved
in[30],
but the methodsof
proof
used here are new and lesscomputational.
The contents ofsections
1, 3,
4 and 5 of this article were announced in various lectures[31 a] [21 a].
Someexamples
ofPoisson-Nijenhuis
structures appear in[31 a].
The connection between solutions of the modifiedYang-Baxter equation
and bihamiltonian structures on the dual of a Liealgebra (see
subsection
6.7),
which is due toMagri,
was announced in a lectureby Magri (E.N.S., Paris, May 1988)
and in[21 a].
1. DEFORMATION OF LIE BRACKETS BY MEANS OF A NIJENHUIS OPERATOR
In this
section,
we construct ahierarchy
ofpairwise compatible
Liebrackets on a Lie
algebra by
means of aNijenhuis
operator, and we prove that theimages
of the center of the Liealgebra
under the powers of theNijenhuis
operator are Abeliansubalgebras
with respect to each of these Liealgebra
brackets.1.1.
Constructing
deformed brackets fromNijenhuis
operators Let E be a vector space over the field of real orcomplex
numbers. Inthe
applications
to differential geometry, E will be the vector space TM of smooth vector fields over M or itscomplexification,
or the vector space T* M of smooth differential 1-forms over M or itscomplexification.
Let Jl
be an E-valued 2-form on E that defines a Liealgebra-structure
on E. The Lie bracket defined
by Jl
will be denotedby [ , ]~
orsimply by
[ , ]
when no confusion ispossible.
Let N be a linear map from E to itself. The
Nijenhuis
torsion[N, N~~
of
N,
with respect to the Liealgebra-structure Jl,
is the E-valued 2-form definedby
for x
and y
in F.DEFINITION 1. 1. -
If
theNijenhuis
torsionof
Nvanishes,
we shall say that N is aNijenhuis
operator.Let denote the E-valued 2-form on E defined
by
for x
and y
in E. The bracket definedby N. ~
will be denotedby [ , ]N .
WThus, by definition,
The various
interpretations
of N . to begiven below,
as well as theexamples
andapplications,
willjustify
the introduction of the bracketAnnales de l’Institut Henri Poincaré - Physique théorique
[ ,
u, which we shallcall,
somewhatimproperly,
thedeformed
bracket.In the sense of the
theory
of deformations of Liealgebras [35],
N . p is atrivial
infinitesimal deformation of
p sincewhere,
for any t inR,
and Jlt
defines a Liealgebra-structure
on E that isisomorphic
to u.In
general, N. Jl
need not define a Liebracket,
evenif Jl
itself does.In
corollary
1. 1 below we prove however thatN. Jl
does define a Liebracket in the
special
case where N is aNijenhuis
operator. We also prove,that p
and are thencompatible
Liealgebra-structures
in thefollowing
sense.
DEFINITION 1. 2. - Two Lie
brackets p and p’
are calledcompatible if
their sum is also a Lie bracket.
If Jl
andj~
arecompatible
Liebrackets,
then for each t in R orC,
Jl +
t Jl’
is a Lie bracket.It follows from the
general theory
of deformations[35]
that ifu’
is aninfinitesimal deformation
of Jl
which is also a Liealgebra-structure,
then rfor each t in [R or
C,
is a Liealgebra-structure
and therefore Jl andJl’
arecompatible.
The one-parameterfamily
of Liealgebra-structures
~ +
t Jl’
is called an actualdeformation
of p, and the Lie brackets Jl + t~’
are the
deformed
brackets. It isonly by
an abuse oflanguage
that we callitself,
when N is aNijenhuis
operator, a deformed Lie bracket.To prove the
properties
of the deformedbrackets,
we shall make use ofthe
Richardson-Nijenhuis
bracket of E-valued forms on E.1.2.
Richardson-Nijenhuis
bracket of E-valued forms on a vector space E and theproperties
of the deformed bracketLet E be a vector space over the field K = ~ or C. Let a be an E-valued K-multilinear form of
degree
a on E. To a therecorresponds
aderivation ia
of
degree a -
I of thegraded algebra
of K-multilinear forms on E with values in anarbitrary
vector spaceF,
defined as follows. For each b-formJ3
with values inF,
ifb >_ 1,
we define thep
withvalues in F
by
and we set
ia (3 = 0
forb = 0,
i. e., whenP
is a function. We have denoted apermutation
of1,
..., a + b -1by
o, and itssignature by
Thex;’s,
i=1,..., a+b-l,
denote elements of E. We remark that theform i03B1 03B2
isdenoted
by P
A a in[ 14] .
Let
[ , ]
denote the commutator in thegraded
sense on the vector space of derivations of the space of E-valued forms on E. The formuladefines a
bracket [ ,
] on the space of E-valued forms on E which coincides with the one definedby Nijenhuis
and Richardson in[35].
We havedenoted this bracket
by [ , ] (small, light
typeface)
todistinguish
it formthe Lie bracket
[ , ] (light
typeface),
the bracket[ , ] (large, light
typeface)
used to denote theNijenhuis torsion,
and from the commutator[ , ] (normal, heavy
typeface).
If a is an E-valued form ofdegree
a, andif P
is an E-valued form of
degree b,
thenThe
Richardson-Nijenhuis
bracket is anticommutative(in
thegraded sense),
and satisfies the Jacobiidentity (in
thegraded sense).
It is known
[35] that Jl
defines a Liealgebra-structure
on E if andonly
if the
Richardson-Nijenhuis
bracket of ~,,Jl],
vanishes.Let N be a linear
mapping
from E toE,
considered as an E-valued 1-form on E.By
the definitionof [N,
yj =iJ!
N -iN
~. _ -N~,
we see thatMoreover,
LEMMA 1. l. - Let
[N,
be theNijenhuis
torsionof N
with respect tothe Lie
algebra-structure y
considered as an E-valued2-form
on E. ThenProof -
Theproof
is astraightforward computation, using
the defini-tion _
for xl, x2, x~ in
E,
wherefdenotes
the sum over thecyclic permutations
over the indices
1,
2 and 3.The deformed structure
N. u
is a Liealgebra-structure
if andonly
if(N .
~., N = o. Now, thepreceding
lemma shows that this condition is satisfied if andonly
if [1.1,[N,
vanishes. Thisquantity
has asimple cohomological interpretation.
Let us set, for any E-valued form a of
degree a,
Annales de l’Institut Henri Poincaré - Physique théorique
It follows from the
graded
Jacobiidentity
for theRichardson-Nijenhuis
bracket that
8~
is acohomology
operator. It is easy to check[35]
that8~
is the
coboundary
operator of the cochaincomplex
of E-valued forms onE,
where E acts on itselfby
theadjoint
action definedby
ji.Clearly,
and
Thus
N. ~
defines a Liealgebra
bracket on E if andonly
if theNijenhuis
torsion of N is a
2-cocycle.
The Lie
algebra-structures y
andN. ~ satisfy
It follows from this fact and from lemma
1. l, that,
when theNijenhuis
torsion of N is a
2-cocycle,
theexpression
[11 + t N . + t N . = + t
(iy,
N. +[N .
J.!, + t2[N.
N. vi vanishesidentically,
for t in R orC,
which expresses the factthat
andare
compatible.
In conclusion:
PROPOSITION 1. l. -
Let
be a Liealgebra-structure
onE,
and let Nbe a linear map
from
E to E. Then N . ~, =ö~
N is a Liealgebra-structure
on E
if
andonly if
theNijenhuis
torsion[N, N]~ of N
is~~-closed,
in whichcase
[, )~
and[, )N . ~
arecompatible.
The
following corollary
will beimportant
in thesequel.
COROLLARY 1. l. - Let ~, be a Lie
algebra-structure
onE,
and let N bea linear map
from
E to E. Assume that theNijenhuis
torsionof
N withrespect to ~u vanishes. Then
(i)
N . ~,defines
a Liebracket,
(ii)
the Lie bracket[, ]N . ~
iscompatible
with the Lie bracket[, )~,
(iii)
N is a Liealgebra-morphism from (E,
N . to(E, y).
Proo, f : - (i)
and(ii)
are aspecial
case ofproposition
1.1.(An
alternateproof
of(i)
will moreover begiven
in section6.)
The last statement followsimmediately
from theidentity
We have shown that
N . ~
is a Lie bracketcompatible with ~
under theassumption
that theNijenhuis
torsion ofN,
with respect to ~ be closed in the Liealgebra-cohomology.
In order to define the iterated brackets and tostudy
theirproperties,
we shall make use of the strongerhypothesis
that the
Nijenhuis
torsion of N vanishes.1.3.
Properties
of the iterated bracketsLet us assume, as in
corollary 1.1,
that N is aNijenhuis
operator. Theidentity
is easy to prove and shows
that,
when the torsion of N with respect to J.lvanishes,
the torsion of N with respect toN. J.l
also vanishes. Therefore the deformationof J.l
intoN . J.l
can beiterated,
and we shall prove belowsome
properties
of the brackets obtained fromN. J.l by
iteration.The
following
lemma is well-known.(See
e. g. note 5 of[26].)
LEMMA 1. 2. -
N~ ~ = o, then, for
eachinteger k >__ I , [Nk,
= o.This property
implies
that we can deform thegiven
Lie bracket notonly by
means of N butby
any power Nk of N. Thefollowing
lemmaexpresses the
associativity
of the deformation process.LEMMA 1. 3 . -
Let J.l
and N be as incorollary
1.1.Then for
eachinteger
Proof. -
Infact,
let ThenIt is clear that
C~==0
and thatI>~ = 2 [N,
Moreover,Therefore,
induction showsthat,
if[N, N~~ = o,
thenO§Q = 0
for allSummarizing
thisdiscussion,
we obtain:PROPOSITION 1.2. -
Let
be a Liealgebra-structure
on E, and let N be aNijenhuis
operator on(E, y).
Let k and m benonnegative integers.
Then
(i) defines
a Lie bracket;(ii)
theNijenhuis
torsionof
Nm with respect to the Liealgebra-structure
Nk . y vanishes;(iii)
the Lie brackets[, )Nk . ~
and[ , ]Nm . ~
arecompatible;
(iv)
Nm is a Liealgebra-morphism from (E,
Nk+m~)
to(E,
Nk.p).
Annales de l’Institut Henri Poincaré - Physique théorique
Proof -
Part(i)
follows from lemma 1. 2 andcorollary
1.1(i).
Inorder to prove
(ii),
we remark that thefollowing
formula is valid for eachinteger ~~0,
or
The
proof
isby computation, using
the result of lemma 1.3. From thisformula,
it follows that if[N, N]~ vanishes,
then the torsion of N with respect to each iterated bracket Nk . J.1 vanishes.Moreover, by
lemma1. 2,
since the torsion of N with respect to the Lie bracketNk .
jnvanishes,
forany
integer m >-_ o,
that of Nm vanishes also.Part
(iii)
follows fromcorollary
1. 1(ii), using
the facts that fork > m, Nk . ~
= N~.(which
follows from lemma1 . 3),
and that Nk - m hasvanishing Nijenhuis
torsion with respect to Nm. y. Part(iv)
follows fromcorollary
1 . 1(iii), using
and[Nm,
= o.1.4. Abelian
subalgebras
We shall now prove an additional property of the deformed Lie brackets which has
applications
in thetheory
ofintegrable
systems.PROPOSITION 1.3. - Let N be a
Nijenhuis
operator on a Liealgebra (E,
and let k and m benonnegative integers.
Then(a)
The centerC~ of (E, y) is
an Abeliansubalgebra of (E,
Nm.J.1).
(b)
Theimage of C~
under Nk is an Abeliansubalgebra of (E, J.1).
Proof -
Let xand y
be in the center of(E, J.1).
Thenwhich proves
(a). Moreover,
since theNijenhuis
torsion of Nk with respectto N"". Jl
vanishes,
which proves
(b).
In
[8],
Caccese states a theorem due to Mishchenko and Fomenko in aform close to the
preceding
result.2. DEFORMATION OF LIE BRACKETS AND HAMILTONIAN SYSTEMS
In this
section,
weexplain
the closerelationship
between twilled Liealgebras
andNijenhuis
operators, and we show how theproperties
ofNijenhuis
operators derived in the first sectionyield,
as acorollary,
involution theorems on the dual of a Lie
algebra. Using Nijenhuis
ope- rators, a class of Hamiltonian systems of Toda type are obtained for eachsemisimple
Liealgebra.
2.1. The
Nijenhuis
operator of a twilled Liealgebra
An operator with
vanishing Nijenhuis
torsion iseasily
constructed on aLie
algebra
g =(E, Jl)
whichsplits,
as a vector space, into a direct sum oftwo Lie
subalgebras,
a and b. The notion of a Liealgebra
which is adirect sum of two Lie
subalgebras
was introduced and studied in[21],
asa
generalization
of Drinfeld’s Liebialgebras,
under the name "twilledextension",
or,preferably,
twilled Liealgebra ("algèbre
de Liebicroisée",
in
French,
see[2]). Shortly
thereafter this notion was defined in[32],
whereit was called a
bicrossproduct
Liealgebra,
and in[28],
where it was calleda double Lie
algebra.
Let
g=a0b,
and let x be theprojection of g
onto aparallel
to b. Forx in g, we shall write x~ for the a-component, x x of x,
and x6
for theb-component,
~20147r~, of x. We shall also write[ , ]
for[ , ]~.
Let N be any
Nijenhuis
operator on the Liealgebra
a suchthat,
for xand y
in g,If N is a
Nijenhuis
operator on a which satisfies(2 . 1 ), denoting
thecanonical
injection
of a into 9by i,
then N = is aNijenhuis
operatoron g. In
fact,
for xand y
in g,The first term vanishes because N is a
Nijenhuis
operator on a, while the second term vanishes because of the additionalrequirement (2.1)
on N,and the last term is zero because b is a Lie
subalgebra
of g. Whence theNijenhuis
torsion of N vanishes.The
identity
of a isobviously
aNijenhuis
operator on a which satisfies condition(2 . 1).
The associatedNijenhuis
operator on g, N = i ° ~, is theAnnales dc l’Institut Henri Poincaré - Physique théorique
projection
of g onto a.Similarly,
theprojection
onto b is aNijenhuis
operator on g.
Nontriviality
for a twilled Liealgebra
means that a and b are bothdifferent
from {0},
andnontriviality
for aNijenhuis
operator means that it is neither invertible norequal
to 0. When a isneither {0 }
nor g, thenthe
projection
of g onto a is a nontrivialNijenhuis
operator on g. Therefore PROPOSITION 2.1. -Any
nontrivial twilled Liealgebra
has a nontrivialNijenhuis
operatorand, therefore,
ahierarchy of deformed
Lie brackets.Let us
remark, however,
that if theNijenhuis
operator is theprojection
of g onto one of the
subalgebras
of a twilled Liealgebra,
thecorresponding
"hierarchy"
hasonly
twoelements, ~
andN.n, since,
in this case, N2 = N.Conversely,
let us show that any finite-dimensional Liealgebra admitting
a
Nijenhuis
operator is a twilled Liealgebra.
This twilled Liealgebra-
structure is nontrivial if the
Nijenhuis
operator is nontrivial.Let E be a finite-dimensional vector space, and let N be a linear
mapping
from E to E. Recall
[17]
that there exists a smallestinteger ~1,
calledthe Riesz index of
N,
such that the sequences of nestedsubspaces
and
both stabilize at rank r.
Thus, by definition,
and
Moreover,
Esplits
as the direct sum of the vector spaces Im Nr andKer Nr,
theFitting
components of N. We shall set a=lm Nr and We now assume thatg = (E, Il)
is a Liealgebra.
We showthat,
if N isa
Nijenhuis
operator, then a and b are Liesubalgebras.
Infact, by
lemma1. 2, [N, N]~ implies [Nr, Thus,
for xand y
in g,Thus
[a, a] c a and,
moreover, since Ker N2 r = KerNr, [b, b]
c b.By
the definition of the Rieszindex,
a = Im Nr is invariant under Nr, and N, the restriction of N~ to a, is invertible. It follows from lemma 1.2 that N is aNijenhuis
operator on a. We showthat,
inaddition,
N satisfiesequation (2 . 1 ).
Since N isinvertible,
it suffices to provethat,
for xand y
in g,
By definition,
and since theNijenhuis
torsion of Npvanishes,
which
expression
isequal
to 0 because Nr vanishes on b. Therefore Nsatisfies
(2 .1 ).
In
conclusion,
nontrivialNijenhuis
operators on a Liealgebra
g are in one-to-onecorrespondence
withpairs (a(j1b, N),
where a(+)b is a nontrivial twilled Liealgebra-structure
on g, and where N is an invertibleNijenhuis
operator on a. In the next
subsection,
we shall consider theidempotent Nijenhuis
operators whichcorrespond
to the case where N is theidentity
of a.
2.2. The N-matrix
approach
to theKostant-Symes
theoremIn this subsection we present "an N-matrix
approach"
to the Kostant-Symes theorem,
to bedistinguished from,
andcompared with,
Semenov-Tian-Shansky’s
"R-matrixapproach"
to thetheory
ofintegrable
sys- tems[41].
Let g be a finite-dimensional Lie
algebra
with Lie bracket[ , ]~ = [ , ].
We consider the dual
g*
of gequipped
with the linear Poisson structuredefined
by y ("Lie-Poisson
structure" or "Kirillov-Kostant-Souriau struc-ture"),
and we denote the Poisson bracket on C~(g*) by { , }
orby { , } g*’ Then, for f and
g inC~g*), and ç
ing*,
Casimir functions are functions on
g*
whose Poisson bracket with any function vanishes.We shall prove that when N is an
idempotent Nijenhuis
operator on(g, y),
i. e., when g is a twilled Liealgebra
and N is theprojection of
g ontoa
factor,
thenis a
Nijenhuis
operator on COO(g*).
We first note thefollowing
facts:and
Annales de l’lnstitut Henri Poincaré - Physique théorique
Proof -
The chain ruleimplies (i).
To prove(ii)
we write the definitions and use thepreceding
result.Since, by assumption,
N2 =N,
we obtainPROPOSITION 2.2. -
If
N is anidempotent Nijenhuis
operator on(g, [, ]J!)’
then % is aNijenhuis
operator on(g*), ~ , } J!).
Proof -
Theproposition
follows from formula(2. 3).
Since % is a
Nijenhuis
operator on Coo(g*),
there is a deformed Poisson bracket of functions ong*,
This bracket is
compatible
with the Lie-Poissonbracket ~ , ~ ~,
but thehierarchy
reduces tojust
two such brackets since JV2 = JV.In the
following proposition,
westudy
the restriction of the Poissonbracket
to the dual of theimage
of N.We consider the
splitting
where a is theimage
of N and b its kernel.Then, N=~7r, where,
asabove, x
is theprojection
of g onto a, and i is the canonicalinjection
of a into g. Themapping ri
is amorphism
of Poisson manifolds from
g*
to a*. Infact,
since i is a Liealgebra- morphism
from a to g, forf and g
in C°~(a*),
Therefore,
forany f and g
in Coo(g*),
Since
and since is the
restriction, of f
toa*,
we see that the Lie- Poisson bracket on a* and thebracket { , } K
ong*
are relatedby
We have
proved:
PROPOSITION 2 . 3. - The linear
mapping
t~c is a Poissonmorphism from
a*
equipped with ~ , ~Q*
tog* equipped
with the Poissonstructure ~ ,
Therefore a* with its canonical Poisson structure is a Poisson sub- manifold of
g* equipped with { , }N.
From(2.4)
andproposition
1 . 3 weobtain
immediately
COROLLARY 2 1
(Kostant-Symes
theorem[22]). - If f
and g are Casimirfunctions
ong*,
thenIn order to compare this "N-matrix
approach"
with the "R-matrixapproach"
ofSemenov-Tian-Shansky,
we first remark that the Liealgebra-
structure { , }j~
ong*
coincides with the Lie-Poisson structure definedby
the deformed Lie bracketN. ~
when restricted to a*. Infact,
Since ç
is ina*,
thenand therefore
proving
that both brackets coincide on a*.Now,
consider the R-bracket ofSemenov-Tian-Shansky,
where
Then,
while
Thus the bracket
[ , ]N . ~
and the bracket[ , ]R
have the sameprojection
onto a.
Thus,
upon restriction toa*,
thebracket { , ~~.
and the Lie-Poisson brackets associated with the deformed Lie bracket[ , ]N . ~
and with the R-bracket[ , ]R
all coincide. In consequence, the Hamiltonian vector fields definedby
these Poisson brackets ong*
coincide on a*.2.3.
Nijenhuis
operators onsemisimple
Liealgebras
and the Toda lattice Let e be asemisimple
real orcomplex
Liealgebra
of rank I. If s isreal,
we assume that it is
split.
We consider a Cartansubalgebra t)
and a system ofsimple
roots (Xi, ..., ~ with respect tot).
Letbe the associated Borel
subalgebra.
Lethi,
...,hi
be a basisof 1). Let xJ
be an
eigenvector
of thesimple
root(x~ ~’=1, ... , I.
The generatorsAnnales de I’Institut Henri Poincaré - Physique théorique