Lie-Butcher series, Geometry, Algebra and Computation

1.1 MB·

by  Hans Z. Munthe-Kaas

Lie–Butcher series,
Geometry, Algebra and Computation
HansZ.Munthe-KaasandKristofferK.Føllesdal
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1 Abstract Lie–Butcher (LB) series are formal power series expressed in terms of
trees and forests. On the geometric side LB-series generalizes classical B-series

] fromEuclideanspacestoLiegroupsandhomogeneousmanifolds.Onthealgebraic
A
side,B-seriesarebasedonpre-LiealgebrasandtheButcher-Connes-KreimerHopf
N
algebra.TheLB-seriesareinsteadbasedonpost-Liealgebrasandtheirenveloping
h. algebras. Over the last decade the algebraic theory of LB-series has matured. The
t purposeofthispaperistwofold.First,weaimatpresentingthealgebraicstructures
a
underlyingLBseriesinaconciseandselfcontainedmanner.Secondly,wereviewa
m
numberofalgebraicoperationsonLB-seriesfoundintheliterature,andreformulate
[
theseasrecursiveformulae.Thisispartofanongoingefforttocreateanextensive

1 software library for computations in LB-series and B-series in the programming
v languageHaskell.
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3 1 Introduction
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.
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ClassicalB-seriesareformalpowerseriesexpressedintermsofrootedtrees(con-
0
nectedgraphswithoutanycycleandadesignatednodecalledtheroot).Thetheory
7
1 hasitsoriginsbacktotheworkofArthurCayley[5]inthe1850s,whereherealized
: that trees could be used to encode information about differential operators. Being
v
forgottenforacentury,thetheorywasrevivedthroughtheeffortsofunderstanding
i
X numericalintegrationalgorithmsbyJohnButcherinthe1960sand’70s[2,3].Ernst
r HairerandGerhardWanner[15]coinedthetermB-seriesforaninfiniteformalse-
a
riesoftheform
H.Z.Munthe-Kaas,K.Føllesdal
Department of Mathematics, University of Bergen, P.O. Box 7803, N-5020 Bergen, e-mail:
[email protected],e-mail:[email protected]
1
2 HansZ.Munthe-KaasandKristofferK.Føllesdal
tτ
B (α,y,t):=y+ ∑ | | a,τ F (τ)(y),
f f
σ(τ)h i
τ T
∈
wherey Rnisa’base’point, f: Rn Rnisagivenvectorfield,T= , , , ,...
∈ → { }
isthesetofrootedtrees, τ isthenumberofnodesinthetree,α: T Raretheco-
| | →
efficientsofagivenseriesand α,τ Rdenotesevaluationofα atτ.Thebracket
h i∈
hintsthatwelaterwanttoconsider α, asalinearfunctionalonthevectorspace
h ·i
spanned by T. The animal Ff(τ): Rn Rn denotes special vector fields, called
→
elementarydifferentials,whichcanbeexpressedintermsofpartialderivativesof f.
Thecoefficientσ(τ) Niscountingthenumberofsymmetriesinagiventree.This
∈
symmetryfactorcouldhavebeensubsumedintoα,butisexplicitlytakenintothe
seriesduetotheunderlyingalgebraicstructures,wherethisfactorcomesnaturally.
TheB-seriest B (α,y,t)canbeinterpretedasacurvestartinginy.Bychoosing
f
7→
differentfunctionsα,onemayencodeboththeanalyticalsolutionofadifferential
equationy(t)= f(y(t))andalsovariousnumericalapproximationsofthesolution.
0
Duringthe1980sand1990sB-seriesevolvedintoanindispensabletoolinanal-
ysisofnumericalintegrationfordifferentialequationsevolvingonRn.Inthemid-
1990sinterestroseintheconstructionofnumericalintegrationonLiegroupsand
manifolds[18,16],andfromthisaneedtointerpretB-seriestypeexpansionsina
differentialgeometriccontext,givingbirthtoLie–Butcherseries(LB-series),which
combinesB-serieswithLieseriesonmanifolds.Itisnecessarytomakesomemodi-
ficationstothedefinitionoftheseriestobeinterpretedgeometricallyonmanifolds:
Wecannotaddapointandatangentvectorasiny+F (τ).Furthermore,itturns
f
•
outtobeveryusefultoregardtheseriesasaTaylor-typeseriesforthemapping
f B , rather than a series development of a curvet B (a,y,t). The target
f f
7→ 7→
spaceof f B isdifferentialoperators,andwecanremoveexplicitreference
f
7→
tothebasepointyfromtheseries.
Themapping f B inputsavectorfieldandoutputsaserieswhichmayrep-
f
• 7→
resent either a vector field or a solution operator (flow map). Flow maps are
expressedasaseriesinhigherorderdifferentialoperators.Wewillseethattrees
encode first order differential operators. Higher order differential operators are
encodedbyproductsoftrees,calledforests.Wewanttoalsoconsiderseriesin
linearcombinationsofforests.
We will in the sequel see that the elementary differential map τ F (τ) is a
f
• 7→
universalarrowinaparticulartypeofalgebras.Theexistenceofsuchauniquely
definedmapexpressesthefactthatthevectorspacespannedbytrees(withcer-
tainalgebraicoperations)isauniversalobjectinthiscategoryofalgebras.Thus
thetreesencodefaithfullythegivenalgebraicstructure.Wewillseethatthealge-
bracomesnaturallyfromthegeometricpropertiesagivenconnection(covariant
derivation) on the manifold. For Lie groups the algebra of the natural connec-
tion is encoded by ordered rooted trees, where the ordering of the branches is
important.Theorderingisrelatedtoanon-vanishingtorsionoftheconnection.
Lie–Butcherseries 3
The symmetry factor σ(τ) in the classical B-series is related to the fact that
•
severaldifferentorderedtreescorrespondtothesameunorderedtree.Thisfactor
isabsentintheLie–Butcherseries.
The time parameter t is not essential for the algebraic properties of the series.
•
SinceFtf(τ)=t|τ|Ff(τ),wecanrecoverthetimefactorthroughthesubstitution
f tf.
7→
WearriveatthedefinitionofanabstractLie–Butcherseriessimplyas
∑ α,ω ω, (1)
h i
ω OF
∈
where
OF= I, , , , , , , ,..., , ,...
{ }
denotes the set of all ordered forests of ordered trees, I is the empty forest, and
α: OF R are the coefficients of the series. This abstract series can be mapped
→
downtoaconcretealgebra(e.g.analgebraofdifferentialoperatorsonamanifold)
byauniversalmappingω F (ω).
f
7→
Wecanidentifythefunctionα: OF Rwithitsseries(1)andsaythataLie–
→
Butcher series α is an element of the graded dual vector space of the free vector
space spanned by the forests of ordered rooted trees. However, to make sense of
this statement, we have to attach algebraic and geometric meaning to the vector
space of ordered forests. This is precisely explained in the sequel, where we see
that the fundamental algebraic structures of this space arise because it is the uni-
versalenvelopingalgebraofafreepost-Liealgebra.Hencewearriveattheprecise
definition:
AnabstractLie–Butcherseriesisanelementofthedualoftheenvelopingalge-
braofthefreepost-Liealgebra.
Wewillinthispaperpresentthebasicgeometricandalgebraicstructuresbehind
LB-seriesinaselfcontainedmanner.Furthermore,animportantgoalforthiswork
istoprepareasoftwarepackageforcomputationsonthesestructures.Forthispur-
posewehavechosentopresentallthealgebraicoperationsbyrecursiveformulae,
ideallysuitedforimplementationinafunctionalprogramminglanguage.Wearein
the process of implementing this package in the Haskell programming language.
The implementation is still at a quite early stage, so a detailed presentation of the
implementationwillbereportedlater.
2 GeometryofLie–Butcherseries
B-seriesandLB-seriescanbothbeviewedasseriesexpansionsinaconnectionon
afibrebundle,whereB-seriesarederivedfromthecanonical(flatandtorsionfree)
connection on Rn and LB-series from a flat connection with constant torsion on a
4 HansZ.Munthe-KaasandKristofferK.Føllesdal
fibre bundle. Rather than pursuing this idea in an abstract general form, we will
provideinsightthroughthediscussionofconcreteandimportantexamples.
2.1 Paralleltransport
LetMbeamanifold,F(M)thesetofsmoothR-valuedscalarfunctionsandX(M)
thesetofrealvectorfieldsonM.Fort Rand f X(M)letΨt,f: M Mdenote
∈ ∈ →
thesolutionoperatorsuchthatthedifferentialequationγ (t)= f(γ(t)),γ(0)=p
0
∈
M has solution γ(t)=Ψt,f(p). For φ F(M) we define pullback along the flow
∈
Ψt∗,f: F(M)→F(M)as
Ψt∗,fφ =φ◦Ψt,f.
Thedirectionalderivative f(φ) F(M)isdefinedas
∈
d
f(φ)= dt Ψt∗,fφ.
(cid:12)t=0
(cid:12)
(cid:12)
Through this, we identify X(M) with the(cid:12)first order derivations of F(M), and we
obtainhigherorderderivationsbyiterating,i.e. f f isthesecondorderderivation
∗
f f(φ):= f(f(φ)). With Iφ =φ being the 0-order identity operator, the set of
∗
all higher order differential operators on F(M) is called the universal enveloping
algebraU(X(M)). This is an algebra with an associative product . The pullback
∗
satisfies
∂
∂tΨt∗,fφ =Ψt∗,ff(φ).
By iteration we find that dn Ψ φ = f(f( f(φ)))= f n(φ) and hence the
dtn t=0 t∗,f ··· ∗
Taylorexpansionofthepullba(cid:12)ckis
(cid:12)
(cid:12)
t2
Ψt∗,fφ =φ+tf(φ)+2!f∗f(φ)+···=exp∗(tf)(φ), (2)
wherewedefinetheexponentialas
∞ tj
exp∗(tf):= ∑ f∗j.
j!
j=0
ThisexponentialisanelementofU(X(M)),ormorecorrectly,sinceitisaninfinite
series,inthecompletionofthisalgebra.WecanrecovertheflowΨt,f fromexp∗(tf)
by letting φ be the coordinate maps. However, some caution must be exercised,
since pullbacks compose contravariantly Ψt,f◦Ψs,g ∗ =Ψs∗,g◦Ψt∗,f, we have that
exp∗(sg) exp∗(tf)correspondstothediffeomorphismΨt,f Ψs,g.
∗ (cid:0) (cid:1) ◦
Numericalintegratorsareconstructedbysamplingavectorfieldinpointsneara
basepoint.Tounderstandthisprocess,weneedtotransportvectorfields.Pullback
ofvectorfieldsis,however,lesscanonicalthanofscalarfunctions.Thedifferential
Lie–Butcherseries 5
geometricconceptofparalleltransportofvectorsisdefinedintermsofaconnection.
AnaffineconnectionisaZ-bilinearmappingB: X(M) X(M) X(M)suchthat
× →
(φf)Bg=φ(fBg)
fB(φg)= f(φ)g+φfBg
forall f,g X(M)andφ F(M).Notethatthestandardnotationforaconnection
∈ ∈
indifferentialgeometryisis∇fg fBg.Ournotationischosentoemphasisethe
≡
operationasabinaryproductonthesetofvectorfields.Thetrianglenotationlooks
nicer when we iterate, such as in (3) below. Furthermore, the triangle notation is
alsostandardinmuchofthealgebraicliteratureonpre-Liealgebras,aswellasin
severalrecentworksonpost-Liealgebras.
Thereisanintimaterelationshipbetweenconnectionsandtheconceptofparal-
leltransport.Foracurveγ(t) M,letΓ(γ)t denoteparalleltransportalongγ(t),
∈ s
meaningthat
Γ(γ)t: TM TM isalinearisomorphismofthetangentspaces.
• s γ(s)→ γ(t)
Γ(γ)s=Id,theidentitymap.
• s
Γ(γ)u Γ(γ)t =Γ(γ)u.
• t ◦ s s
Γ dependssmoothlyons,t andγ.
•
FromΓ,letusconsidertheactionofparalleltransportpullbackofvectorfields,for
t∈Rand f ∈X(M)wedenoteΨt∗,f: X(M)→X(M)theoperation
Ψt∗,fg(p):=Γ(γ)t0g(γ(t)), forthecurveγ(t)=Ψt,f(p).
Anyconnectioncanbeobtainedfromaparalleltransportastherateofchangeofthe
paralleltransportpullback.ForagivenΓ wecandefineacorrespondingconnection
as
d
fBg:= dt Ψt∗,fg.
(cid:12)t=0
(cid:12)
Conversely, we can recoverΓ from B by(cid:12)solving a differential equation. We seek
(cid:12)
a power series expansion of the parallel transport pullback. Just like the case of
scalars,itholdsalsoforpullbackofvectorfieldsthat
∂
∂tΨt∗,fg=Ψt∗,ffBg,
henceweobtainthefollowingTaylorseriesofthepullback
t2 t3
Ψt∗,fg=g+tfBg+ 2 fB(fBg))+3!fB(fB(fBg)))+···. (3)
Recallthatinthecaseofpullbackofascalarfunction,weused f(g(φ))=(f g)(φ)
∗
toexpressthepull-backintermsofexp (tf).Whetherornotwecandosimilarlyfor
∗
vectorfieldsdependsongeometricpropertiesoftheconnection.Wewouldliketo
extendBfromX(M)toU(X(M))suchthat fB(gBh)=(f g)Bhandhence(3)
∗
6 HansZ.Munthe-KaasandKristofferK.Føllesdal
becomesΨt∗,fg=exp∗(tf)Bg. However, this requires that f B(gBh)−gB(f B
h)= f,g Bh,where f,g := f g g f istheJacobibracketofvectorfields.
∗ − ∗
ThecurvaturetensoroftheconnectionR: X(M) X(M) End(X(M))isdefined
∧ →
as J K J K
R(f,g)h:= fB(gBh) gB(fBh) f,g Bh.
− −
Thus,weonlyexpecttofindasuitableextensionofBtoU(X(M))ifBisflat,i.e.
whenR=0. J K
In addition to the curvature, the other important tensor related to a connection
isthetorsion.GivenB,wedefineanF(M)-bilinearmapping : X(M) X(M)
· × →
U(X(M))as
f g:= f g fBg. (4)
· ∗ −
Theskew-symmetrisationofthisproductcalledthetorsion
T(f,g):=g f f g X(M),
· − · ∈
andif f g=g f wesaythatBistorsionfree.
· ·
The standard connection on Rn is flat and torsion free. In this case the algebra
X(M),B forms a pre-Lie algebra (defined below). This gives rise to classical
{ }
B-series. More generally, transport by left or right multiplication on a Lie group
yieldsaflatconnectionwheretheproduct isassociative,butnotcommutative.The
·
resulting algebra is called post-Lie and the series are called Lie–Butcher series. A
thirdimportantexampleistheLevi–Civitaconnectiononasymmetricspace,where
isaJordanproduct,T =0andRisconstant,non-zero.Thisthirdcaseisthesubject
·
offorthcomingpapers,butwillnotbediscussedhere.
2.2 TheflatCartanconnectiononaLiegroup
LetGbeaLiegroupwithLiealgebrag.ForV gandp GweletVg:=TR V
p
∈ ∈ ∈
T G. There is a 1–1 correspondence between functions f C∞(G,g) and vector
p
∈
fieldsξ X(G)givenasξ (p)= f(p)p.Leftmultiplicationwithq Ggivesrise
f f
∈ ∈
toaparalleltransport
Γ : T G T G:Vp Vqp.
q p qp
→ 7→
Thistransportisindependentofthepathbetween pandqpandhencegivesriseto
a flat connection. We express the corresponding parallel transport pullback on the
spaceC∞(G,g)as
(Γq∗f)(p)= f(qp)
whichyieldstheflatconnection
d
(fBg)(q)= g(exp(tf(q))q).
dt
(cid:12)t=0
(cid:12)
(cid:12)
Thetorsionisgivenas[22] (cid:12)
Lie–Butcherseries 7
T(f,g)(p)= [f(p),g(p)] .
g
−
Thetwooperations fBgand[f,g]:= T(f,g)turnC∞(G,g)intoapost-Liealge-
−
bra,seeDefinition3below.ThisisthefoundationofLie–Butcherseries.
WecanalternativelyexpresstheconnectionandtorsiononX(G)viaabasis E
j
{ }
forg.Let∂ X(G)betherightinvariantvectorfield∂ (p)=E p.ForF,G X(G),
j j j
∈ ∈
whereF= fi∂,G=gj∂ and fi,gj F(G),wehave
i j
∈
FBG= fi∂i(gj)∂j
F G= figj∂∂
i j
·
T(F,G)= figj(∂∂ ∂ ∂).
i j j i
−
WereturntoBdefinedonC∞(G,g).LetU(g)bethespanofthebasis{Ej1Ej2···Ejk},
whereE E E U(g)correspondstotherightinvariantk-thorderdifferential
j1 j2··· jk ∈
operator∂ ∂ ∂ U(X(G)).OnU(g)wehavetwodifferentassociativeprod-
jk··· j2 j1 ∈
ucts, the composition of differential operators f g and the ’concatenation prod-
∗
uct’ f g = f g f Bg which is computed as the concatenation of the basis,
· ∗ −
fiE gjE = figjEE . The general relationship between these two products and
i j i j
·
BextendedtoU(g)isgivenin(28)–(31)below.Inparticularwehave
fB(gBh)=(f g)Bh,
∗
whichyieldstheexponentialformoftheparalleltransport
Ψt∗,fg=exp∗(tf)Bg,
whereexp (tf)isgivingustheexactflowof f.
∗
Wecanalsoformtheexponentialwithrespecttotheotherproduct,
t2 t3
exp·(tf)=I+tf+ f f+ f f f+ .
2 · 3! · · ···
Whatisthegeometricmeaningofthis?Wesaythatavectorfieldgisparallelalong
f iftheparalleltransportpullbackofgalongtheflowof f isconstant,andwesay
that g is absolutely parallel if it is constant under any parallel transport. Infinites-
imally we have that g is parallel along f if f Bg=0 and g is absolutely parallel
if f Bg=0 for all f. InC∞(G,g) the absolutely parallel functions are constants
g(p)=V, which correspond to right invariant vector fields ξ X(G) given as
g
∈
ξ (p)=Vp.Theflowofparallelvectorfieldsarethegeodesicsoftheconnection.
g
Ifgisabsolutelyparallel,wehaveg g=g g+gBg=g g,andmoregenerally
∗ · ·
gn =gn,henceexp (g)=exp(g).If f(p)=g(p)atapoint p G,thentheyde-
∗ · ∗ ·
∈
finethesametangentatthepoint.Hence fn(p)=gn(p)foralln,andweconclude
· ·
that exp(f)(p)=exp(g)(p)=exp (g)(p). Thus, the concatenation exponential
· · ∗
exp(f)ofageneralvectorfield f producestheflowwhichinagivenpointfollows
·
thegeodesictangentto f atthegivenpoint.
8 HansZ.Munthe-KaasandKristofferK.Føllesdal
On a Lie group, we have for two arbitrary vector fields represented by general
functions f,g C∞(G,g)that
∈
(exp·(tf)Bg)(p)=g(exp(tf(p))p). (5)
2.3 Numericalintegration
Lie–Butcherseriesanditscousinsaregeneralmathematicaltoolswithapplications
innumerics,stochasticsandrenormalisation.Theproblemofnumericalintegration
onmanifoldsisaparticularapplicationwhichhasbeenanimportantsourceofin-
spiration.Wediscussasimpleillustrativeexample.
Example1(Lie–trapezoidal method). Consider the classical trapezoidal method.
For a differential equation y0(t)= f(y(t)), y(0)=y0 on Rn a step from t =0 to
t=hisgivenas
h
K= (f(y )+f(y ))
0 1
2
y =y +K.
1 0
Consider a curve y(t) G evolving on a Lie group such that y(t)= f(y(t))y(t),
0
∈
where f C∞(G g)andy(0)=y .IntheLie-trapezoidalintegratorastepfrom
0
∈ →
y toy y(h)isgivenas
0 1
≈
h
K= (f(y )+f(y ))
0 1
2
y =exp (K)y ,
1 g 0
whereexp : g GistheclassicalLiegroupexponential.Wecanwritethemethod
g →
asamappingΦtrap: X(M) Diff(G)fromvectorfieldstodiffeomorphismsonG,
→
givenintermsofparalleltransportonX(M)as
1
K= (f+exp·(K)Bf) (6)
2
Φtrap(f):=exp·(K). (7)
Tosimplify,wehaveremovedthetimesteph,butthiscanberecoveredbythesub-
stitution f hf.NotethatwepresentthisasaprocessinU(X(M)),withoutaref-
7→
erencetoagivenbasepointy0.ThemethodcomputesadiffeomorphismΦtrap(f),
which can be evaluated on a given base point y . This absence of an explicit base
0
pointfacilitatesaninterpretationofthemethodasaprocessintheenvelopingalge-
braofafreepost-Liealgebra,anabstractmodelofU(X(M))tobediscussedinthe
sequel.
Lie–Butcherseries 9
Abasicproblemofnumericalintegrationistounderstandinwhatsenseanumer-
icalmethodΦ(tf)approximatestheexactflowexp (tf).Theorderoftheapproxi-
∗
mationiscomputedbycomparingtheLB-seriesexpansionofΦ(tf)andexp (tf),
∗
andcomparingtowhichorderint thetwoseriesagree.
Thebackwarderror ofthemethodisdefinedasamodifiedvectorfield f such
h
that the exact flowof f interpolates the numericalsolution at integer times1. The
h
combinatorialdefinitionofthebackwarderroris e
e
exp∗(fh)=Φ(hf).
The backward error is an important teool which yields important structural infor-
mation of the numerical flow operator f Φ(hf). The backward error analysis
7→
is fundamental in the study of geometric properties of numerical integration algo-
rithms[9,14].
Yetanotherproblemisthenumericaltechniqueofprocessingavectorfield,i.e.
we seek a modified vector field f such that Φ(f )=exp (f). An important tool
h h ∗
intheanalysisofthistechniqueisthecharacterizationofasubstitutionlaw.What
happenstotheseriesexpansionofeΦ(hf)if f isreeplacedbyamodifiedvectorfield
f expressedintermsofaseriesexpansioninvolving f?
h
The purpose of this essay is not to pursue a detailed discussion of numerical
aenalysis of integration schemes. Instead we want to introduce the algebraic struc-
turesneededtoformalizethestructureoftheseriesexpansions.Inparticularwewill
presentrecursiveformulasforthebasicalgebraicoperationssuitableforcomputer
implementations.
We finally remark that numerical integrators are typically defined as families
of mappings, given in terms of unspecified coefficients. For example the Runge–
Kutta family of integrators can be defined in terms of real coefficients a s
{ i,j}i,j=1
and b s as
{ j}j=1
s
Ki=exp·(∑ai,jKj)Bf, fori=1, ,s
···
j=1
s
ΦRK(f)=exp·(∑bjKj).
j=1
In a computer package for computing with LB-series we want the possibility of
computingseriesexpansionsofsuchparametrizedfamilieswithoutspecifyingthe
coefficients.Thisisaccomplishedbydefiningthealgebraicstructuresnotoverthe
concrete field of real numbers R, but instead allowing this to be replaced by an
abstractcommutativringwithunit,suchase.g.theringofallrealpolynomialsin
theindeterminates a s and b s .
{ i,j}i,j=1 { j}j=1
1Technicalissuesaboutdivergenceofthebackwarderrorvectorfieldisdiscussedin[1].
10 HansZ.Munthe-KaasandKristofferK.Føllesdal
3 AlgebraicstructuresofLie–Butchertheory
We give a concise summary of the basic algebraic structures behind Lie–Butcher
series.
3.1 Algebras
Allvectorspacesweconsiderareoverafield2kofcharacteristic0,e.g.k R,C .
∈{ }
Definition1(Algebra). An algebra A, is a vector space A with a k-bilinear
{ ∗}
operation : A A A.A iscalledunitalifithasaunitIsuchthatx I=I x
∗ × → ∗ ∗
forallx A.The(minus-)associatoroftheproductisdefinedas
∈
a (x,y,z):=x (y z) (x y) z.
∗ ∗ ∗ − ∗ ∗
Iftheassociatoris0,thealgebraiscalledassociative.
Definition2(Liealgebra).ALie-algebraisanalgebra g,[, ] suchthat
{ · ·}
[x,y]= [y,x]
−
[[x,y],z]+[[y,z],x]+[[z,x],y]=0.
Thebracket[, ]iscalledthecommutatororLiebracket.Anassociativealgebra
· ·
A, giverisetoaLiealgebraLie(A),where[x,y]=x y y x.
{ ∗} ∗ − ∗
A connection on a fibre bundle which is flat and with constant torsion satisfies
the algebraic conditions of a post-Lie algebra [22]. This algebraic structure first
appearedinapurelyoperadicsettingin[27].
Definition3(Post-Liealgebra).Apost-Liealgebra P,[, ],B isaLiealgebra
{ · · }
P,[, ] togetherwithabilinearoperationB: P P P suchthat
{ · ·} × →
xB[y,z]=[xBy,z]+[x,yBz] (8)
[x,y]Bz=aB(x,y,z)−aB(y,x,z). (9)
Apost-LiealgebradefinesarelationshipbetweentwoLiealgebras[22].
Lemma1.Forapost-LiealgebraP thebi-linearoperation
x,y =xBy yBx+[x,y] (10)
−
definesanotherLiebracket.
J K
Thus,wehavetwoLiealgebrasg= P,[, ] andg= P, , relatedbyB.
{ · ·} { · · }
2Inthecomputerimplementationswearerelaxingthistoallowkmoregenerallytobeacommu-
J K
tativering,suchase.g.polynomialsinasetofindeterminates.Inthislattercasethek-vectorspace
shouldinsteadbecalledafreek-module.Wewillnotpursuethisdetailinthisexposition.