Serre presentations of Lie superalgebras

0.33 MB·

by  R. B. Zhang

SERRE PRESENTATIONS OF LIE SUPERALGEBRAS
R.B.ZHANG
1
1
0 ABSTRACT. AnanalogueofSerre’stheoremisestablishedforfinitedimensionalsim-
2 pleLiesuperalgebras,whichdescribespresentationsintermsofChevalleygenerators

n andSerretyperelationsrelativetoallpossiblechoicesofBorelsubalgebras.Theproof
a ofthetheoremisconceptuallytransparent;italsoprovidesanalternativeapproachto
J
Serre’stheoremforordinaryLiealgebras.

7
1

]
T
1. INTRODUCTION
R
. 1.0.1. AwellknowntheoremofSerre gavepresentationsoffinitedimensionalsemi-
h
t simpleLiealgebrasintermsofChevalleygeneratorsandSerrerelations. Itwasgener-
a
m alisedtoKac-MoodyalgebraswithsymmetrisableCartanmatricesbyGabberandKac
[9]. The theorem and its generalisation now provide the standard method to present
[

simpleLiealgebras andKac-Moodyalgebras [14],as wellas theassociated quantised
1
v universalenvelopingalgebras[4, 12].
4 A natural question is how to present simple contragredient Lie superalgebras (i.e.,
1
Lie superalgebras with Cartan matrices) in a similar way. Surprisingly this was only
1
3 seriously studied after quantised universal enveloping superalgebras [2] had become
.
1 popular in the early 90s because of their applications in a variety of areas such as low
0 dimensional topology [20, 29], statistical physics [2] and noncommutative geometry
1
[22, 30, 31].
1
: In the Lie superalgebra setting, unconventional higher order relations [19] are re-
v
i quiredbesidetheusualSerrerelations,andtheiroriginissomewhatmysterious. Since
X
a Serre type presentation is always given relative to a chosen Borel subalgebra, the
r
a issueisfurthercomplicatedbythefact[13,14]thatasimplecontragredientLiesuper-
algebraadmitsclassesofBorel subalgebras,whichare notWeylgroup conjugate.
1.0.2. At thepresent, investigationon Serre typepresentationsforLie superalgebras
is still rather incomplete even in the finite dimensional case. Presentations relative to
many non-distinguishedBorel subalgebras of such Lie superalgebras have neverbeen
constructed(seeRemark3.4). ThecrucialquestiononwhethertheSerretyperelations
obtained so far are complete (i.e., whether they are all the defining relations needed
Date:January12,2011.
2010MathematicsSubjectClassification. Primary17B05;Secondary17B20,17B22,17B10.
Keywordsandphrases. Liesuperalgebras,rootsystems,presentations.
SupportedbytheAustralianResearchCouncil.
1
2 R.B.ZHANG
for the Lie superalgebras under consideration) has not been answered satisfactorily.
Therefore, there is the need of a systematic treatment of Serre presentations for the
finitedimensionalsimplecontragredientLiesuperalgebras,andthispaperaimstopro-
videsuchatreatment.
1.0.3. It was Leites and Serganova [19] who first obtained the higher order Serre
relations for sl relative to the so-called distinguished Borel subalgebra (for which
m|n
the simpleroots are the easiest to describe). The corresponding quantum relations for
U (sl ) wereconstructedin[24, 5]. Yamane[26]wrotedownhigherorderquantum
q m|n
Serre relations for quantised universal enveloping superalgebras of finite dimensional
simpleLiesuperalgebrasforthedistinguishedandsome(butnotall)non-distinguished
Borelsubalgebras. Intheensuingyears,muchfurtherworkwasdonetofindSerretype
relations for Lie superalgebras by Leites and collaborators [6, 7, 1] and by Yamane
[27].
References [6, 7] and [26, 27] represent the current state of the problem of con-
structing Serre type presentations for the finite dimensionalsimple contragredient Lie
superalgebras. [Reference [27] is largely on affine superalgebras.] However, the pa-
pers [26, 27] left out presentations of exceptional simple Lie superalgebras relative to
non-distinguishedBorel subalgebras. Reference [6]inprincipletreated alltheDynkin
diagrams which could potentially require higher order Serre relations, but the rela-
tions in [6] and [26, 27] look very different and it is not clear at all whether they are
equivalent.
1.0.4. The problem on whether the Serre type relations constructed were complete
wasonlyinvestigatedbycomputercalculations. Accordingto[6,§1],completenessof
the relationsof [6]was verified by computers for finitedimensionalsimplecontragre-
dientLiesuperalgebras,butaconceptualproofislacking. Theproblemisopenforthe
Serre typerelationsgivenin [26, 27], and soisalso intheinfinitedimensionalcase.
We comment that in the cases considered in [26], completeness of the relations
can in principle be deduced from the existence of a non-degenerate invariant bilinear
form between the quantised universal enveloping superalgebras of the upper and low
triangular Borel subalgebras, by using Geer’s result [10] that quantised universal en-
veloping superalgebras are trivial deformations. However, it is a highly complicated
mattertoestablishthenon-degeneracyofthebilinearformeveninthecaseofordinary
quantised universal enveloping algebras (see, e.g., [21]). Many of the representation
theoretical results required for proving the non-degeneracy are lacking for quantised
universalenvelopingsuperalgebras,rendering thesupercasemuch moredifficult.
1.0.5. In thispaper, we givea completetreatment of the Serre presentations of finite
dimensional simple contragredient Lie superalgebras, proving an analogue of Serre’s
theorem relative to all possible choices of Borel subalgebras. Comparing our results
withthoseof[26](intheq→1limit),wehavemanymorehigherorderSerrerelations
whicharenecessary,especiallyinthecaseofexceptionalLiesuperalgebrasrelativeto
SERREPRESENTATIONSOFLIESUPERALGEBRAS 3
non-distinguished Borel subalgebras. Our method is also different from those in the
literature. It in particularautomaticallyshowsthecompletenessoftherelationswhich
weconstruct.
1.0.6. Letusnowdescribemorepreciselytheresultsofthispaper. Givenarealisation
oftheCartan matrixA=(a )ofasimplecontragredientLiesuperalgebrawiththeset
ij
ofsimplerootsP ={a ,...,a },weintroduceanauxiliaryLiesuperalgebrag˜,which
b 1 r
is generated by Chevalley generators {e, f , h | i = 1,2,...,r} subject to quadratic
i i i
relations only (see Definition 3.1, where more informative notation is used). Let r be
theZ -gradedmaximalidealofg˜ thatintersectstriviallytheCartansubalgebraspanned
2
by all h . Then L := g˜/r is the simple Lie superalgebra which we started with in all
i
cases except intypeA(n,n)whereL issl (seeTheorem3.3).
n+1|n+1
We introducea Z -graded ideal s of theauxiliary Liesuperalgebra, which is gener-
2
ated by explicitly given generators. A main result proved in Theorem 3.10 states that
s = r, or equivalently, g := g˜/s ∼= L. From this result, we deduce a super analogue
of Serre’s theorem, Theorem 3.11, which givespresentations ofthe finitedimensional
simplecontragredientLiesuperalgebrasrelativetoallpossiblechoicesofBorelsubal-
gebras.
ThecompletenessoftherelationsinTheorem 3.11isguaranteed by Theorem 3.10.
1.0.7. Theproof ofTheorem 3.10 makes use ofa Z-grading of g˜, which descends to
L and g to give Z-gradings to these Lie superalgebras. Write L=⊕ L and g=⊕ g
k k k k
with respect the Z-gradings. Lemma 3.8 states that L ∼=g as Lie superalgebras and
0 0
∼
L =g asg -modulesforallk6=0. ThenTheorem 3.10followsfrom thislemma.
k k 0
The unconventional Serre relations can now be understood as arising from two
sources: the conditions for g to be irreducible g -modules; and the requirement
±1 0
that [g ,g ]=L and similarrequirementsat otherdegrees.
±1 ±1 ±2
Recall that Yamane [27] used odd reflections [25] to find such relations. Leites and
collaborators [19, 6] usedhomologicalalgebratechniquesand deduced relations from
certain spectral sequences.
The approach developed here is quite different from the methods in [6, 7, 1] and in
[26, 27] at both the conceptual and technical level. It has the advantage of automati-
callygeneratingacompletesetofrelationsthatisminimal. Conceptuallytheapproach
isquitetransparentinthesensethatonecanseehowthedefiningrelationsarise. Italso
providesanalternativeapproachtoSerre’stheoremforfinitedimensionalsemi-simple
Liealgebras, seeRemark 5.2.
We also note that the proof in [9] of the generalised Serre theorem for Kac-Moody
algebras with symmetrisable Cartan matrices relied on structural properties of Verma
modules such as theirembeddings,and also made useof thequadratic Casimiropera-
tor. Theauthorsofboth[27]and[6]commentedonobstaclesingeneralisingtheproof
to Lie superalgebras, especially difficulties related to the quadratic Casimir operator.
4 R.B.ZHANG
We may also add that one no longer has the properties of (generalised) Verma mod-
ules required by [9] in the context of Lie superalgebras, and this appears to be a more
seriousdifficulty.
1.0.8. The organisation of the paper is as follows. Section 2 reviews Kac’s classifi-
cation of finite dimensional simple classical Lie superalgebras [13], and also clarifies
certain subtlepoints about Cartan matrices and Dynkin diagrams in this context. Sec-
tion 3 contains the statements of the main results, Theorem 3.10 and Theorem 3.11,
whichgivepresentationsofcontragredientLiesuperalgebrasinarbitraryrootsystems.
The proof of Theorem 3.10, which implies Theorem 3.11 as a corollary, is given by
using the key lemma, Lemma 3.8. Sections 4 and 5 are devoted to the proof of the
key lemma. An outline of the proof is given in Section 4.2 to explain its conceptual
aspects. We end the paper with a discussion of possible generalisation of the method
developed here to affine Kac-Moody superalgebras to construct Serre type presenta-
tionsinSection 6.
Two appendices are also included. Appendix A gives the root systems and Dynkin
diagramsofallsimplecontragredientLiesuperalgebras[13,8,3]. Thematerialisused
throughoutthepaper, and is also necessary in orderto makeprecise thedescriptionof
Dynkin diagrams in non-distinguished root systems. Appendix B describes the struc-
ture of some generalised Verma modules of lowest weight type and their irreducible
quotients,which entertheproofofLemma3.8.
Acknowledgement. Iwishto thankProfessorDimitryLeitesforhelpfulsuggestions.
2. FINITE DIMENSIONAL SIMPLE LIE SUPERALGEBRAS
Inthissection,wepresentsomebackgroundmaterial,andclarifysometrickypoints
aboutCartan matricesand DynkindiagramsofLiesuperalgebras.
2.1. Finite dimensional simple Lie superalgebras. We work over the field C of
complexnumbersthroughoutthepaper.
2.1.1. Classification. A Lie superalgebra g is a Z -graded vector space g = g ⊕g
2 0¯ 1¯
endowed with a bilinear map [ , ]:g×g−→g, (X,Y)7→[X,Y], called the Lie super-
bracket, which is homogeneous of degree 0, graded skew-symmetric and satisfies the
super Jacobian identity. The even subspace g of a Lie superalgebra g=g ⊕g is a
0¯ 0¯ 1¯
Lie algebra in its own right, which is called the even subalgebra of g. The odd sub-
space g forms a g -module under the restriction of the adjoint action defined by the
1¯ 0¯
Liesuperbracket. Ifg isareductiveLiealgebraandg isasemi-simpleg -module,g
0¯ 1¯ 0¯
is calledclassical[13, 23].
TheclassificationofthefinitedimensionalsimpleLiesuperalgebraswascompleted
in the late70s. Thetheorem belowis taken from [13], which is stillthe best reference
on Lie superalgebras. Historical information and further references on the classifica-
tioncan befound in[16, 17](also see[23]).
SERREPRESENTATIONSOFLIESUPERALGEBRAS 5
Theorem 2.1. The finite dimensional simple classical Lie superalgebras comprise of
thesimplecontragredientLiesuperalgebras
A(m,n), B(0,n), B(m,n), m>0, C(n), n>2, D(m,n), m>1,
F(4), G(3), D(2,1;a ), a ∈C\{0,−1},
and simplestrangeLiesuperalgebrasP(n)andQ(n)(n≥1).
ThesimplecontragredientLiesuperalgebrasadmitnon-degenerateinvariantbilinear
forms, while the strange Lie superalgebras P(n) and Q(n) do not. In the remainder of
thepaper, weshallconsideronlycontragredientsimpleLiesuperalgebras.
The A, B, C and D series are essentially the special linear and orthosymplectic Lie
superalgebras, which are familiarexamples ofLie superalgebras. Theexceptional Lie
superalgebras F(4),G(3) and D(2,1;a ) are less well-known, but one can understand
theirstructuresgiventhedescriptionoftheirrootsin AppendixA.1.
Let g = g ⊕g be a simple contragredient Lie superalgebra, and choose a Cartan
0¯ 1¯
subalgebra h for g, which by definition is just a Cartan subalgebra of g . Denote
0¯
by ga the root space of the root a , and call a even (resp. odd) if ga ⊂ g0¯ (resp.
ga ⊂g1¯). Denoteby D 0 and D 1 thesets oftheeven and odd rootsrespectively,and set
D =D ∪D . Let(, ):h∗×h∗ −→CdenotetheWeylgroupinvariantnon-degenerate
0 1
symmetric bilinear form on h∗, where the Weyl group of g is by definition the Weyl
groupofg . Arootb willbecalledisotropicif(b ,b )=0. Notethatallisotropicroots
0¯
are odd.
A Borel subalgebra of g is a maximal soluble Lie super subalgebra containing a
Borel subalgebraof g . A new feature in the present context is that Borel subalgebras
0¯
are not always conjugate under the Weyl groups. All the conjugacy classes of Borel
subalgebras were given in [13, pp. 51-52] [14, Proposition 1.2]. In particular, Kac
described a particularly convenient Borel subalgebra, which he called distinguished,
for each simple contragredient Lie superalgebra. We shall call a root system with the
set of simpleroots determined by this Borel subalgebra the distinguishedroot system.
In thiscase, thereexistsonlyoneodd simpleroot.
2.1.2. Cartan matrices and Dynkin diagrams. The precise forms of the Cartan ma-
trices and Dynkin diagrams will be crucial in Section 3. However, there do not exist
canonical definitions for them in the Lie superalgebra setting, thus we spell out the
detailsofourdefinitionshere.
Let P ={a ,a ,...,a } be the set of simple roots of a simple contragrediant Lie
b 1 2 r
superalgebragrelativetoaBorelsubalgebrab. TheCartanmatrixandDynkindiagram
provide a convenient way to describe P . We define a Cartan matrix in the following
b
way. Denote by Q ⊂{1,2,...,r} the subset such that a ∈D for all t ∈Q . Let l2 be
t 1 m
theminimumof|(b ,b )|forall non-isotropicb ∈D ifg6=D(2,1;a ). Ifg isD(2,1;a ),
letl2 betheminimumofall|(b ,b )|>0(b ∈D ),whichareindependentofthearbitrary
m
6 R.B.ZHANG
parametera . Let
0, ifgisoftypeB, (a i,a i), if(a ,a )6=0,
k = d = 2 i i
(cid:26) 1, otherwise; i ( 2lm2k , if(a i,a i)=0.
Introducethematrices
B=(b )r , b =(a ,a ),
ij i,j=1 ij i j
D=diag(d ,...,d ),
1 r
then theCartan matrixA associatedtotheset ofsimplerootsP is defined by
b
(2.1) A=D−1B.
When it is necessary to indicate the dependence on Q , we write (A,Q ) for the Cartan
matrix.
Notethatifa isnon-isotropic,a = 2(a i,a t) isanon-positiveintegerforallt. How-
i it (a i,a i)
ever,ifa isisotropic,thena = 2(a ,a )canbeanintegerofanysignorzero(except
t tj l2 t j
m
in typeD(2,1;a )). Ifb 6=0,wedefine
ij
(2.2) sgn =signofb .
ij ij
As weshallseeinSection2.2, thesesignsprovidetheadditionalinformationrequired
to recoveraCartan matrixfromits Dynkindiagram.
Remark 2.2. OurdefinitionoftheCartan matrixdiffersfrom theusualoneduetoKac
[13]. In Kac’s definition, if b = 0, then d =(a ,a ) for the smallest k such that
ss s s s+k
d 6=0. Notethatin ourdefinition,noneofthesignssgn is lost.
s ij
TheDynkindiagramassociatedwith(A,Q )consistsofrnodes,whichareconnected
bylines. Thei-thnodeiscolouredwhiteifi6∈Q ,blackifi∈Q buta isnotisotropic,
i
and greyifa isisotropic.
i
If(A,Q )isoftypeD(2,1;a ),theDynkindiagramisobtainedbysimplyconnecting
thei-thand j-th nodesby onelineifa 6=0and writeb at theline.
ij ij
In allothercases, wejointhei-thand j-thnodes byn lines,where
ij
n =max(|a |,|a |), ifa +a ≥2;
ij ij ji ii jj
n =|a |, ifa =a =0.
ij ij ii jj
When the i-th and j-th nodes are not both grey, say, the i-th one is not grey, and
connectedbymorethanonelines,wedrawanarrowpointingtothe j-thnodeif−a =
ij
1 and pointingtothei-thnodeif−a >1.
ij
The Dynkin diagrams of the simple contragredient Lie superalgebras are given in
thetablesin AppendixA.2.
SERREPRESENTATIONSOFLIESUPERALGEBRAS 7
2.2. Comments on Dynkin diagrams. From the Cartan matrices in our definition,
one can recover the corresponding root systems. Dynkin diagrams also uniquely rep-
resent Cartan matrices, except in the cases of osp and sl . The Dynkin diagrams
4|2 2|2
of these superalgebras relative to the distinguished root systems are exactly the same,
butthetwoLiesuperalgebras arenon-isomorphic.
This problem can be resolved by incorporating the signs sgn into the Dynkin dia-
ij
gram, e.g., by placing sgn at the line(s)connecting two grey nodes i and j. Then the
ij
modifiedDynkindiagram arerespectivelygivenby
sl : − + osp : − −
(2.3) 2|2 i y i, 4|2 i y i.
As weshallsee, thesignsentertheconstructionofhigherorderSerre relations.
In this paper we did not include the additional information of these signs in the
definition of Dynkin diagrams, as they would make the diagrams look cumbersome.
Also,thereis noambiguityaboutthesignsinall theotherDynkindiagrams.
Similarsignswerealso discussedin[27].
Recallthatifweremoveasubsetofvertices(i.e.,nodes)andalltheedgesconnected
to these vertices from a Dynkin diagram of a semi-simple Lie algebra, we obtain the
Dynkin diagram of another semi-simple Lie algebra of a smaller rank. This corre-
sponds to taking regular subalgebras. In the context Lie superalgebras, the notion of
regularsubalgebrasstillexists,butsomeexplanationisrequiredatthelevelofDynkin
diagrams.
Definition 2.3. Call asub-diagramG ′ ofaDynkindiagramG fullifforany twonodes
i and j in G ′, the edges between them in G , the arrows on the edges, and also the b
ij
labels oftheedges when G isoftypeD(2,1;a ),are allpresent inG ′.
ConsiderforexampletheDynkindiagram
(cid:0) y
(cid:0)
(cid:0)(cid:0)
i > y
@
@ y
ofF(4),which hasthefollowingfullsub-diagramsbesideothers:
i > y y, y y.
(2.4)
Notethatnoneoftheseappears in Tables1 and 2.
The reason is that the sub-matrices in the Cartan matrix of F(4) associated with
thesefullsub-diagramsarenotCartan matricesinthestrictsense. Theproblemliesin
the definitionof a when the nodei is grey, which involvesthe numberl . The l for
ij m m
F(4) is not the correct ones for the full sub-diagrams. By properly renormalising the
bilinearformsontheweightspacesassociatedwiththem,thefullsub-diagramscanbe
cast intotheform
8 R.B.ZHANG
i y y, y y,
which arerespectivelyDynkindiagramsforsl and sl .
3|1 2|1
WecalltheDynkindiagramsinTable1andTable2standard,andtheoneslikethose
in (2.4)non-standard.
We mention that if a Lie superalgebra g is contained as a regular subalgebra in
another Lie superalgebra, defining relations of g can in principle be extracted from
relationsofthelatterbyconsideringsub-diagramsofDynkindiagrams. However,this
involves subtleties, as we have just discussed, and requires more care than hitherto
exercisedin theliterature.
3. PRESENTATIONS OF LIE SUPERALGEBRAS
Inthissection,wegeneraliseSerre’stheoremforsemi-simpleLiealgebrastocontra-
gredient Lie superalgebras, obtaining presentations for the Lie superalgebras in terms
ofChevalleygeneratorsand defining relations.
3.1. AnauxiliaryLiesuperalgebra. WestartbydefininganauxiliaryLiesuperalge-
brafollowingthestrategyof[15]. Let(A,Q )withA=(a )r betheCartanmatrixof
ij i,j=1
oneofthesimplecontragredientLiesuperalgebrasrelativetoagivenBorelsubalgebra
b. Let P betheset ofsimpleroots relativeto thisBorel subalgebra.
b
Definition3.1. Letg˜(A,Q )betheLiesuperalgebrageneratedbyhomogeneousgener-
ators e, f ,h (i=1,2,...,r), wheree , f foralls∈Q areodd whiletherest areeven,
i i i s s
subjectto thefollowingrelations
[h ,h ]=0,
i j
(3.1) [h ,e ]=a e , [h, f ]=−a f ,
i j ij j i j ij j
[e, f ]=d h , ∀i, j.
i j ij i
Let n˜+ (resp. n˜−) be the subalgebra generated by all e (resp. all f ) subject to the
i i
relevantrelations,andh=⊕r Ch, theCartan subalgebra. Thenitiswellknownand
i=1 i
easy to prove (following the reasoning of [15, §1]) that g˜(A,Q )= n˜+⊕h⊕n˜−. The
Lie superalgebra is graded g˜(A,Q )=⊕n ∈Qg˜n by Q=ZP b, with g˜0 =h. Note hat n˜+n
(rep. n˜−−n )iszero unlessn ∈QN, whereN={1,2,...} andQN =NP b, thatis,
(3.2) n˜+ =⊕n ∈QNn˜+n , n˜− =⊕n ∈QNn˜−−n .
Let r(A,Q ) be the maximal Z -graded ideal of g˜(A,Q ) that intersects h trivially.
2
Set r± =r(A,Q )∩n˜±. Then r(A,Q )=r+⊕r−. The following fact follows from the
maximalityofr(A,Q ).
Lemma 3.2. Let S =S +∪S − with S ± ⊂n˜± be a subset of g˜(A,Q ) consisting of ho-
mogeneouselements. If [f ,S +]⊂CS + and [e,S −]⊂CS − foralli,then S ⊂r(A,Q ).
i i
SERREPRESENTATIONSOFLIESUPERALGEBRAS 9
Proof. The given conditions on S imply that the ideal generated by r(A,Q )∪S inter-
sects h trivially,hencemustbeequal tor(A,Q )by themaximalityofthelatter. (cid:3)
In particular, if X± ∈ n˜± satisfy [f ,X+] = 0, and [e,X−] = 0 for all i, then they
i i
belongto n˜± respectively.
Let usdefinetheLiesuperalgebra
g˜(A,Q )
L(A,Q ):= .
r(A,Q )
Wehavethefollowingresult.
Theorem 3.3. Let g be a finite dimensional simple contragredient Lie superalge-
bra, and let (A,Q ) be the Cartan matrix of g relative to a given Borel subalgebra.
Then L(A,Q ) is isomorphic to g unless g= A(n,n), and in the latter case L(A,Q ) ∼=
sl .
n+1|n+1
Proof. This follows from Kac’s classification [13] of the simple contragredient Lie
superalgebras (see Theorem 2.1) except in the case of A(n,n). In the latter case, we
have detA= 0. Therefore, L(A,Q ) contains a 1-dimensional center, and the quotient
ofL(A,Q )by thecenterisA(n,n). Hence L(A,Q )isisomorphictosl . (cid:3)
n+1|n+1
3.2. Maintheorem.
3.2.1. StandardandhigherorderSerreelements. Letusfirstdefinesomeelementsof
g˜(A,Q ), which will play a crucial role in studying the presentation of Lie superalge-
bras.
Call thefollowingelementsthestandardSerreelements:
(ad )1−aij(e ), (ad )1−aij(f ), fori6= j, witha 6=0 ora =0;
ei j fi j ii ij
[e ,e ], [f , f ], fora =0.
s s s s ss
We also introduce higher order Serre elements if the Dynkin diagram of (A,Q ) con-
tainsfull sub-diagramsofthefollowingkind:
j t k
(1) × y × with sgnjtsgntk = −1, the associated higher order Serre ele-
mentsare
[e ,[e ,[e ,e ]]], [f ,[f ,[f , f ]]];
t j t k t j t k
j t k
(2) × y > i, theassociatedhigherorderSerre elementsare
[e ,[e ,[e ,e ]]], [f ,[f ,[f , f ]]];
t j t k t j t k
j t k
(3) × y > y, theassociatedhigherorderSerre elementsare
[e ,[e ,[e ,e ]]], [f ,[f ,[f , f ]]];
t j t k t j t k
10 R.B.ZHANG
j t k
(4) y y< i,theassociatedhigherorderSerre elementsare
[[e ,e ],[[e ,e ],[e ,e ]]],
j t j t t k
[[f , f ],[[f , f ],[f , f ]]];
j t j t t k
i j t k
(5) × i y< i, theassociated higherorderSerre elementsare
[[e,[e ,e ]],[[e ,e ],[e ,e ]]],
i j t j t t k
[[f ,[f , f ]],[[f , f ],[f , f ]]];
i j t j t t k
yt
(cid:0)
i(cid:0)
(6) × , theassociatedhigherorderSerre elementsare
@
@ ys [et,[es,ei]]−[es,[et,ei]],
[f ,[f , f ]]−[f ,[f , f ]];
t s i s t i
1 2 3 4
(7) i > y< i i, which is a Dynkin diagram of F(4), the associated
higherorderSerre elementsare
[E,[E,[e ,[e ,e ]]]],
2 3 4
[F,[F,[f ,[f , f ]]]],
2 3 4
where E =[[e ,e ],[e ,e ]]and F =[[f , f ],[f , f ]];
1 2 2 3 1 2 2 3
1 2 3 4
(8) i > y i< i, which is a Dynkin diagram of F(4), the associated
higherorderSerre elementsare
[[e ,e ],[[e ,e ],[e ,e ]]−[[e ,e ],[[e ,e ],[e ,e ]],
1 2 2 3 3 4 2 3 1 2 3 4
[[f , f ],[[f , f ],[f , f ]]−[[f , f ],[[f , f ],[f , f ]];
1 2 2 3 3 4 2 3 1 2 3 4
k t j
(9) i > y y, which only appears in Dynkin diagrams of F(4), the as-
sociated higherorderSerre elementsare
[e ,[e ,[e ,e ]]],
t j t k
[f ,[f ,[f , f ]]];
t j t k
(cid:0)yj
i
(cid:0)
(10) y , whichonlyappears inoneoftheDynkindiagramsofF(4),
@@
@@yk
theassociatedhigherorderSerre elementsare
2[e,[e ,e ]]+3[e ,[e ,e]],
i k j j k i
2[f ,[f , f ]]+3[f ,[f , f ]];
i k j j k i