Canonical endomorphism field on a Lie algebra

0.18 MB·English

by  Jerzy Kocik

Canonical endomorphism field
on a Lie algebra
2
1
0
2 JerzyKocik

n DepartmentofMathematics,SIU,Carbondale,IL62901,USA
a
E-mail:[email protected]
J

6

] Abstract
h
p WeshowthateveryLiealgebraisequippedwithanatural(1,1)-variant
- tensorfield,the“canonicalendomorphismfield”,naturallydeterminedby
h
theLiestructure,andsatisfyingacertainNijenhuisbracketcondition.This
t
a observationmaybeconsideredascomplementarytotheKirillov-Kostant-
m Souriautheoremonsymplecticgeometryofcoadjointorbits.Weshowits
relevanceforclassicalmechanics,inparticularforLaxequations.Weshow
[
thatthespaceofLaxvectorfieldsisclosedunderLiebracketandweintro-

1 duceanewbracketforvectorfieldsonaLiealgebra.Thisbracketdefines
v anewLiestructureonthespaceofvectorfields.
4
2000MSC:17B08,53C15,53C80,70G45,70G60,70H03,70H05.
5
3 Keywords:Liealgebra,Nijenhuisbracket,classicalmechanics,Laxequa-
1 tions, Kirillov-Kostant-Souriau theorem, Poisson theorem, graphical lan-
1. guage.
0
2
Notation. Weshalldistinguishbetweenpurelyalgebraicanddifferentialprod-
1
uctsbyusingtwotypesofbrackets:
:
v [[ , ]] —Liealgebraproduct,
i
X [ , ] —Liecommutatorofvectorfields,Schoutenbracket,Nijenhuisbracket.
r The summation convention over repeated indices is adopted throughout the
a
paper.
1 Introduction
Itiswell-knownthattheunderlyingdualspaceL∗ofaLiealgebraLpossesses
—asamanifold—acanonicalPoissonstructureintermsofasmoothbi-vector
field Ω ∈ ∧2TL∗, which satisfies the Jacobi condition [Ω,Ω] = 0, and, when
restrictedto coadjoint orbits, is nondegenerate and therefore invertible into a
symplecticstructure[16,17,12].Theexistenceofthesesymplecticsheetsisthe
contentoftheKirillov-Kostant-SouriauTheorem[3,9,15].
1
InthispaperwepresentanoverlookedfactthattheLiealgebraLitselfalso
possesses—asamanifold—anaturaldifferential-geometricobject,namelya
(1,1)-typetensor field A ∈ T(1,1)L thatwe shallcallthe canonicalendomor-
phism field on L. The principal geometric property of A is that it is propor-
tionaltoitsownNijenhuisderivative(Theorem2.1).
We discussthe relevanceofthis objectfordynamicalsystems. Itturns out
thatwhatHamiltonequationsareforthedualspaceL∗,Laxequationsarefor
L.TheprincipalpropertyofAassuresthatthespaceof “Laxvectorfields” is
closed under the Lie commutator and, moreover, it allows one tointroduce a
newbracketofvectorfieldsonL,whichistheanalogforLaxequationsofthe
PoissonbracketonHamiltonianvectorfields.
2 The canonical endomorphism field on a Lie alge-
bra
Customarilyone definesa Liealgebraasa linearspaceLwith aproductL×
L7→Ldenoted[[v,w]](doublebracket).Theproductisbilinear,skew-symmetric
(i),andsatisfiestheJacobiidentity(ii):
(i) [[v,w]]=−[[w,v]]
(1)
(ii) [[v,[[w,z]]]]+[[w,[[z,x]]]]+[[x,[[w,z]]]]=0
Inabasis{e },thecommutatorcanberepresentedvia“structureconstants”:
i
[[e ,e ]]=cke (2)
i j ij k
Hereweshallratherfollow[10]anddefineaLiealgebraasapair{L, c}where
cisa(1,2)-typetensorthatintheabovebasisis
c= 1ck εi∧εj ⊗e (3)
2 ij k
where {εi} is the dual basis. The algebra product becomes a secondary, de-
rived, concept: [[v, w]] = (v ∧ w) c = i i c. Similarly, the adjoint action of
w v
v ∈ Lisdefinedsimply asa(1,1)-tensorad = v cinL.Of course,fromthe
v
structuralpointofviewbothdefinitionsareequivalent,{L,c}≡{L, [[·, ·]]}.
ThepointofthepresentpaperistolookatthespaceLasaflatmanifoldand
considervariousdifferential-geometricobjectsonit.(WeshallassumethatLis
arealandfinite dimensional.)The linearstructureofthismanifold allowone
toprolonganytensorT inLtothe(”constant”)tensorfieldT onthemanifold
e
L.Inparticular,the manifold Lisequippedwith aconstant(1,2)-typetensor
fieldλ=c:
e λ= 1ck dxi∧dxj ⊗∂ (4)
2 ij k
where{xi}arecoordinatesonLassociatedwiththe basis{e }andwherewe
i
denote∂ ≡∂/∂xi.ThemanifoldLisalsoequippedwithanaturalvectorfield,
i
theLiouvillevectorfield,whichinalinearcoordinatesystemis
J =xi∂ (5)
i
2
Hereisourbasicobservation:
Theorem2.1. ThemanifoldoftheLiealgebraLpossessesanaturalfieldofendomor-
phisms(i.e.,a(1,1)-varianttensorfield)A∈T(1,1)Ldefinedby
A=J λ (6)
ItsNijenhuisderivative[A,A]isavector-valuedbiform
[A,A]=−2λ A (7)
MoreoverAactsontheadjointorbitsonL.
WeshallcallAthecanonicalendomorphismfieldonL.Inthecoordinate
description,AanditsNijenhuisderivativeare
A = xick dxj ⊗∂
ij k
(8)
[A,A] = −xk ci cp (dxa∧dxb)⊗∂
kp ab i
TheendomorphismfieldAmaybeviewedasafamilyoflocaltransformations
thatatpointx∈LcanberepresentedbymatrixAk(x)=xick.
j ij
Beforewegiveitsproof,letusrestatethetheoreminmorestandardterms.
The natural isomorphism of a tangent space at any x ∈ L with the space L
itselfwillbedenotedbyµ :T L→L.ThenTheorem2.1statesthateveryLie
x x
algebraLpossesses, asamanifold,aunique naturaltensorfieldA ∈ T(1,1)L,
which atpoint x ∈ L isdefined asanendomorphism taking atangentvector
v ∈T Lto
x
Ax(v)=(µ−x1 ◦adx ◦µx)(v) (9)
or,inasomewhatsloppynotation,A(v)=[[x,v]].ItsNijenhuisderivative[A,A]
isavector-valuedbiformtheevaluationofwhichequalsforanyv,w ∈TL:
[A,A](v,w)=−2A([[v,w]])
(10)
=−2([[Av,w]]+[[v,Aw]])
atpointx∈L,thedependenceofwhichwassuppressedinthenotation.
Remark2.2. The canonicalendomorphism field A isdefined for anarbitrary
algebraanditsdifferential-geometricproperties,includingtheNijenhuisbracket
[A,A],willreflectthetypeofthisalgebra.InthepresentpaperwerestricttoLie
algebraswheretheJacobiidentityimpliesparticularlypleasantconsequences.
The above theorem may be viewed as a counterpart of the KKS theorem:
the essence of which is that the dual space L∗ is equipped with a bi-vector
field Ω = x ck∂i ∧∂j (in our language Ω = J λ). Instead of the Nijenhuis
k ij
bracketwehavetheSchoutenbracket[Ω, Ω] = 0.ThusΩdefinesaPoisson
Sch
structure,which,moreover,restrictstothecoadjointorbits,onwhichitsinverse
ωdefinesasymplecticstructure,ω =0.Section8summarizestheseparallels.
3
3 Lie algebra in pictures
Tensor calculus gains much transparency when expressed in graphical lan-
guage.
Basic Glyphs.Herearethebasicglyphscorrespondingtovarioustensors:
s v α A g
T
scalar vector 1-form endomorphism scalarproduct (3,2)-variant
tensor
wheresisascalar,v isavector,αisacovector,Aisanendomorphism, g isa
metricorbiform.Thelinkswitharrows andlinkswithcircles rep-
resent the contravariant and the covariant attributes of a tensor, respectively.
You may think of them as contravariant/covariant (upper/lower) indices in
somebasisdescription.Scalarshavenone.
The “in” and “out” links may go any direction. Turning and weaving in
space doesnot have any meaning (unlike in some other convections). For in-
stance:
thisrepresentation T isasgoodasthis T
The links may leave the box at any position, but the order of the point of
departureis fixed: the contravariant indices are ordered clockwise, while the
covariantindicescounterclockwise.Linksmaycrosswithoutanymeaningim-
plied.
Glyphs may be composed into pictograms that represent terms resulting by
manipulation with tensors. The tensor contractions are obtained by joining
”ins”with”outs”.Herearesomebasiccases:
Evaluation:Hereistheevaluationofacovectoronaform:
α α
hα,vi≡α(v)= orsimply
v v
4
Scalarproduct:Thescalarproductof twovectorsisascalarg(v, w) =,butif
onlyonevectoriscontractedwithg,thentheresultisaone-form:
g g g
g(v, w)= ∼= g(v, ·)≡v g =
v w v w v
EndomorphismAactingonavectorvorcovectorαresultsinavectororcov-
ector,respectively:
α
A
v → Av= α → A∗α=
A
v
Trace may be represented by connecting “in’ with “out” in a pictogram; If
A,B,C ∈EndLareendomorphismofsomelinearspaceL,thenwehave:
TrA= A , TrAB = A B , TrABC = A B C
Thenotablepropertyoftraceofacompositionofendomorphisms, namelyits
invariance under cyclic permutation of the entries, TrA1 ◦ ... ◦ Ak−1 ◦ Ak =
TrA2 ◦...◦Ak ◦A1,becomesingraphicallanguageverifiablewithasimplicity
ofamantraonajapamala.
Lie algebra in pictures. An algebra is defined by a (1,2)-variant tensor c, as
shownbelowontheleft.Alsoaproductandadjointrepresentationisshown:
Algstructure= c [[v, w]]= c ad = c
v
v w v
Ifasinglealgebraisconsidered,theletter“c”willbesuppressed.
InthecaseofaLiealgebra,besidesskew-symmetrywehavetheJacobiiden-
5
tity,whichmaybewrittenthisway
= − (11)
a b b a
a b
Thelabelsaandbareonlytodiscernbetweendifferententries.
Perhaps the simplest derived object is a characteristic one-form χ ∈ L∗ the
valueofwhichonavectorv ∈Lisχ(v)=−Trad .Itspictographis
v
χ =
(Thisone-formvanishesforsemisimplealgebras.)
TheKillingform isdefinedasaninnerproductK(v, w) = Trad ad .Inthe
v w
diagrammaticscriptitiseasytodefinethecorresponding2-covarianttensorK
K =
Every Lie algebra possesses a skew-symmetric exterior Lie 3-form ω that for
any triple v,w,z ∈ L takes value ω(v,w,z) = Trad ad . Using diagram-
[[v,w]] z
matic script we may “draw”the form ω directly — here it is, simplified with
theuseofJacobiidentity(11):
ω = = −
a b b a
a b
whereaandbaremerelylabelstodistinguishthecovariantentries.Ifweuse
symbolthe∧or“alt”insidealooptodenotethesignedsumoverallpermuta-
tionsofentriesofatensor(skewsymmetrization),thentheLie3-covariantform
is
6
alt
ω ≡ 1
3
4 Differential geometry on a Lie algebra
LetusnowlookatthedifferentialgeometryofLiealgebraviewedasamani-
fold.InthediagrammaticlanguagetheobjectsofTheorem2.1are
Definition: A= Theorem: 1[A,A]=
2
J J
Since the contraction with J is introduces dependence on poisition (coordi-
natesx),weshalluserathernotationthatwillbeeasierperceptually.Thus,for
instance:
A= ≡
J x
Every element (vector) v ∈ L defines a “constant” vector field v ∈ XL
onmanifoldLobtainedbyparalleltransport;incoordinates,,ifv = veie then
i
v = vi∂ .ThecanonicalendomorphismfieldAonmanifoldLappliedtosuch
i
fieelds definesa representationof Lie algebra L in terms of vector fields on L,
namelywitheveryalgebraelementv ∈L,weassociateavectorfield
X =Av
v
e (12)
=xivjck∂ =
ij k
x v
Proposition4.1. Themapv →X definesahomomorphism{L, [[·, ·]]}→{XL,[·, ·]}
v
(theinfinitesimalrepresentationofLintermsofXL):
[X ,X ]=X (13)
v w [[v,w]]
IfthecenterofListrivial,themappresentsamonomorphism.
Proof. ThepropositionreadilyfollowsfromtheJacobiidentity.
7
Corollary4.2. Thefollowingareconvenientformulae
^
(i) [X , w]= [[v,w]]
v
e
^
(ii) [[v,w]]= [[v,w]]
e e
(iii) [v,w]=0
e e
Theimageof AspansateverypointasubspaceofthetangentspaceofL,
defininginthiswayadistribution
D =ImA=span{X |v ∈L} (14)
v
Theintegralmanifoldsofthisdistributioncoincidewiththeadjointorbitsde-
termined by the action of a Lie group on Lie algebra. Note however that we
may define “adjoint orbits” without reference to the Lie group simply as the
integralmanifoldsOofD,satisfyingTO=D.
Nowweprovethetheorem.
ProofofTheorem2.1: RecallthattheNijenhuisbracket[K,K]ofavector-valued
one-form (endomorphism field) K with itself is a vector-valuedbi-form that,
evaluatedontwofieldsX andY,takesthevalueaccordingto
1
[K,K](X,Y)=[KX,KY]−K[KX,Y]−K[X,KY]+K2[X,Y] (15)
2
(see, e.g., [13]). Evaluating (half of) the Nijenhuis bracket [A, A] on two con-
stantvectorfieldsv andw andusingformulaeofProposition4.1and4.2,one
e e
gets
1
[A,A](v,w)=[X ,X ]−A([X ,w])−A([v,X ])+0
v w v w
2 e e e e
=X −X −X
[[v,w]] [[v,w]] [[v,w]]
=−X
[[v,w]]
Inparticular,substitutionX = ∂ andY = ∂ leadstothecoordinateformula
a b
(8).Now,letusshowthatAcanberestrictedtoorbits,i.e.,A(T O) ⊂ T Ofor
x x
eachpointx∈O.First,rewrite(9)forX ∈T L:
x
A(X)=µ−1([[x,(µ (X)]])
x x
VectorofthevectorfieldX atpointx∈Lcanbeexpressedas
v
X (x)=A(v)=µ−1([[x,v]])
v x
Thus
A(X )=µ−1([[x,µ (X )]])=µ−1([[x,[[x,v]]]]) =X ∈T O
v x x v x [[x,v]] x
whichwastobeproven.
8
Example4.3. Considerthe2-dimensionalsolvablealgebradefinedby[e ,e ]=
1 2
e .Then
2
A=x dx ⊗∂
1 2 2
[A,A]=−2x (dx ∧dx )⊗∂
1 1 2 2
Theadjoint orbitsarelinesparalleltoe andthe canonicalendomorphism —
2
whenrestrictedtoanyofthem—becomesadilation.
Example 4.4. The Lie algebra of 3-dimensional rotations, so , is defined by
3
relations]]e ,e [[=ε e .Thus
i j ijk k
A=x (dx2⊗∂ −dx3⊗∂ )+(cyclicterms)
1 3 2
[A,A]=dx1∧dx2⊗(x1∂ −x2∂ )+(cyclicterms)
2 1
TheorbitsarespheresdefinedbytheKilling form.Ontheunitsphere,tensor
Aformsanalmostcomplexstructure,A◦A=−id.
Remark 4.5. Although the Nijenhuis bracket(7) vanishes for two-step nilpo-
tentalgebras,(includingHeisenberg-typealgebras[8,7]),ingeneralitdoesnot,
andthereforeendomorphismfieldAisingeneralnotintegrable.Notethatfor
vectorfields ofinfinitesimal representation,the bi-form(7) takesatanypoint
xavector-value
[A,A](Xv,Xw)=µ−x1 ◦[[x,[[[[x,v]],[[x,v]]]]]] (16)
ThusArestrictedtoanorbitO ⊂Lis(locally)integrableif[[x,[[[[x,v]],[[x,v]]]]]] =
0 for every x ∈ O and every v,w ∈ L. This is true for so(n), n ≤ 4 and for
nilpotentalgebrasoftheupper-triangularn×nmatrices,n≤5.
5 Other basic properties of the endomorphism field
Thefundamentalpropertyofthecanonicalendomorphismfield(Theorem2.1)
is
[A,A]=−2λ A
OtherbasicpropertiesofthegeometryofaLiealgebraaresummarizedbelow:
Corollary5.1. TheendomorphismfieldonaLiealgebrasatisfies:
(i) £ A=A
J
(ii) J A=0
(17)
(iii) ImA(cid:12) ∼=Imad2
(cid:12)O x
(iv) KerA(cid:12)(cid:12)O ∼=Keradx∩Imadx
Hereisapropertyanalogoustothecoadjointrepresentationpreservingthe
Kirillov-PoissonstructureonthedualLiealgebra.
9
Proposition5.2. TheendomorphismAispreservedbytheactionoftheadjointrep-
resentation,thatis
£ A=0 ∀v ∈L (18)
Xv
Proof. UseLeibnizruletoshowthat(£ A)(w) = 0foreveryw:(£ A)(w) =
Xv Xv
£ (A(w))−A£ w =£ X −A[[v,w]]=X −X =0.
Xv Xv Xv w [[v,w]] [[v,w]]
Proposition5.3. TheendomorphismfieldonaLiealgebrasatisfies:
(i) Tr(A◦A)=K(J,J)
(ii) Tr(A)=χ(J) (19)
(iii) K(Av,w)=−K(v,Aw)
where the objects are as follows: K is the Killing form defined for two vectors as
K(v,w) = Tradv ◦ adw. When evaluated for (J,J), it becomes a quadratic scalar
functionK(J,J) = xaxbck ci . Similarly,χ ∈ L∗ is a characteristicform onL de-
ai bk
finedχ(v)=Trad .Property(iii)statesthattheendomorphismAisskew-symmetric
v
withrespecttotheKilling(possiblydegenerated)scalarproduct.
The endomorphism definesfor everyk = 1,2,...,a scalarfunctionof the
powertrace
Ik =TrAk =Tr(adx ◦ ... ◦adx) (20)
that will be called Casimir polynomials on L. In the diagrammatical language
theyare:
I = I = I = I =
1 2 3 4
x x x x x x x x x x
etc.Clearly,thesecondinvariantisaquadraticfunctionrelatedtoKillingform
and will be denoted κ = I = K(J,J) = κ, but the third is obviously not
2
relatedtotheLie3-form.
Corollary 5.4. Differentials of the trace functions are among the annihilators of A,
i.e.,
A dI =0 (21)
k
10