Cohomological properties of unimodular six dimensional solvable Lie algebras

0.24 MB·English

by  Maura Macrì

COHOMOLOGICAL PROPERTIES OF UNIMODULAR SIX
DIMENSIONAL SOLVABLE LIE ALGEBRAS
2
1 MAURAMACR`I
0
2
Abstract. InthepresentpaperwestudysixdimensionalsolvableLiealgebras
n with special emphasis on those admitting a symplectic structure. We list
a all the symplectic structures that they admit and we compute their Betti
J
numbersfindingsomepropertiesaboutthecodimensionofthenilradical. Next,

0 we consider the conjecture of Guan about step of nilpotency of a symplectic
2 solvmanifoldfindingthatitistrueforallsixdimensionalunimodularsolvable
Liealgebras. Finally,weconsidersomecohomologiesforsymplecticmanifolds

] introduced byTsengand Yauinthe context of symplecticHogde theory and
G
we use them to determine some six dimensional solvmanifolds for which the
D HardLefschetzpropertyholds.
.
h
t
a
m Introduction
[ A solvmanifold M = G/Γ is a compact homogeneous space of a solvable Lie

2 group G, i.e. a compact quotient of a solvable Lie group G by a lattice Γ. A
v special class of solvmanifolds, called nilmanifolds, was introduced by Malcev [10]
8 and corresponds to the particular case when G is a nilpotent Lie group.
5
Both classesof manifolds have been particularly importantfor producing exam-
9
5 ples of compact symplectic manifolds which do not admit any K¨ahler structure.
. Hence the study of the topology of solvmanifolds (and in particular their coho-
1
mological properties) is particularly interesting, especially when they are endowed
1
1 withasymplecticstructure. Inthiscontext,theHardLefschetzproperty,formality
1 and symplectic Hodge theory play an important role (see for instance [3]).
:
v Hence one needsto compute the de Rhamcohomologyofa solvmanifold,which,
i in some situations can be done using invariant differential forms, i.e. by the
X
Chevalley-Eilenberg cohomology H∗(g) of g. This is the case if the Mostow con-
r
a dition holds, namely, the algebraicclosures A(AdG(G)) and A(AdG(Γ)) are equal,
[11]and[14,Corollary7.29]. Specialinstancesareprovidedbynilmanifolds[13]and
completely solvable Lie groups G [7], i.e., the adjoint representation ad : g → g
X
have only real eigenvalues for all X ∈g.
Unlike nilpotent Lie groups there is no simple criterion to understand whether
a solvable Lie group G admits a lattice, but a necessary condition is that G is
unimodular, i.e., for all X ∈g, trad =0, where g is the Lie algebra of G.
X
Nilpotent Lie algebras and solvable Lie algebras have been classified up to di-
mension 5 (see for instance [1]). In dimension 6, the number of possible solvable
Lie algebras is very large. A complete classification can be obtained by using the
results in the papers [1, 12, 16]. In this paper we improve some classifications,
considering only the case of six dimensional solvable unimodular Lie algebras.
2000 Mathematics Subject Classification. 53C30;17B30.
1
SIX DIMENSIONAL LIE ALGEBRAS 2
Solvmanifolds up to dimension six admitting an invariant symplectic structure
(invariant means that it comes from a symplectic form on the Lie algebra) were
studied by Bock [1]. In particular, he considered the conditions of being cohomo-
logically symplectic, formality and Hard Lefschetz property.
Inthepresentpaperwestudysixdimensionalunimodularsolvable(non-nilpotent)
Lie algebras with special emphasis on those admitting a symplectic structure. Six
dimensional nilpotent Lie algebras admitting symplectic structures were classified
in [15].
InSection1welistallthesymplecticstructuresthatsixdimensionalunimodular
solvable (non-nilpotent) Lie algebras admit (Table 3) and we consider the conjec-
ture of Guan [6] about steps of a symplectic solvmanifold [6], namely that if a
solvmanifoldG/Γ admits a symplectic structure then G is at most 3-step solvable.
We find that this is true for a six dimensional Lie group whose Lie algebra is uni-
modular,indeed this holds for all six dimensional unimodular solvable Lie algebra,
also those which do not admit symplectic structures, (see Proposition 1).
In Section 2 we compute their Betti numbers (i.e., the dimensions of their
Chevalley-Eilenberg cohomology) finding some properties about the codimension
ofthe nilradical. Recallthatthe nilradical ofaLiealgebragisits largestnilpotent
ideal.
Namely, in Section 2 we prove as main result
Theorem 1. Let g be a six dimensional unimodular, solvable, non-nilpotent Lie
algebra
• if it admits a symplectic structure, then it has positive, non zero, second
Betti number, i.e., b (g)>0.
2
• if b (g)=1, then its nilradical has codimension 1 and b (g)=0 if and only
1 2
if b (g)=0.
3
• ifitsnilradicalhascodimensiongreaterthen1,thenb (g)≥2andb (g)=1
1 2
if and only if b (g)=0.
3
The Betti numbers of the 6-dimensionalLie algebras with 5-dimensionalnilrad-
ical were also computed by M. Freibert and F. F. Schulte-Hengesbach [4].
Finally, we consider the Hard Lefschetz property and some cohomologies for
symplecticmanifoldsintroducedbyTsengandYau[18]inthecontextofsymplectic
Hogde theory. In particular, we show that these cohomologies can be computed
using invariant forms, provided this is the case for the Rham cohomology (see
Theorem3 in Section 3). We apply this result, together with the list of symplectic
structures on solvable Lie algebras (Table 3), to show that some solvmanifolds
satisfy the Hard Lefschetz property (Theorem 4).
In the Appendix we include two Tables. In Table 2 we list all the solvable, non-
nilpotentunimodularsixdimensionalLiealgebras,indicatingthedifferentialofthe
generators of the dual algebra. This list is based on the classifications given in [1],
[12] and [16] and has the same notation for the Lie algebras. In Table 3 we list all
the symplectic structures on six dimentional solvable unimodular Lie algebras.
Acknowledgements. I would like to thank Antonio Otal and Anna Fino for the
precious help given in the last version of the paper.
SIX DIMENSIONAL LIE ALGEBRAS 3
1. Symplectic Structures
LetgbearealLiealgebraofdimension2n. Werecallthatasymplectic structure
ong is a closed2-formω in g∗ such that ωn 6=0. Let g be a six dimensionalreal
solvableunimodular Lie algeVbraandlet {X1,...,X6} be an orderedbasis ofg, then
a 2-formω is associatedin a natural way to a matrix M =(ω )∈M(6,R), where
ij
ω :=ω(X ,X ), and ωn 6=0⇔detM 6=0.
ij i j
By direct computation we prove
Theorem 2. The six dimensional real solvable, non-nilpotent unimodular Lie al-
gebras admitting a symplectic structure are the following:
g0,−1=(−26,−36,0,−46,56,0),
6.3
g0,0 =(−26,−36,0,−56,46,0),
6.10
g21,−1,0 =(−23+ 1.16,−1.26,36,−46,0,0),
6.13 2 2
g−1,21,0 =(−23+ 1.16,26,−1.36,−46,0,0),
6.13 2 2
g−1 =(−23,−26,36,−26−46,−36+56,0),
6.15
g−1,−1=(−23,26,−36,−36−46,56,0),
6.18
g0 =(−23,0,−26,−46,56,0),
6.21
g0,0,ε =(−23−ε.56,0,−26,−36,0,0), ε6=0
6.23
g0,0,ε =(−23−ε.56,0,0,−36,−46,0),
6.29
g0,0 =(−23,0,−26,56,−46,0),
6.36
g0 =(−23,36,−26,−26+56,−36−46,0),
6.38
g0,−1=(−35−16,−45+26,−36,46,0,0),
6.54
g0,0 =(−35+26,−45−16,46,−36,0,0),
6.70
g =(−25+16,−45,−24−36−46,−46,56,0),
6.78
g0,±1,−1 =(−25+16,15+26,∓45−36,±35−46,0,0),
6.118
n±1 =(−45,−15−36,−14+26∓56,56,−46,0),
6.84
gp,−p,−1⊕R=(−15,−p.25,p.35,45,0,0),
5.7
g−1⊕R=(−25,0,−35,45,0,0),
5.8
g0 ⊕R=(−25,0,−45,35,0,0),
5.14
g0,0,r⊕R=(−25,15,−r.45,r.35,0,0),
5.17
gp,−p,±1⊕R=(−p.15−25,15−p.25,p.35∓45,±35+p.45,0,0),
5.17
g0,0,±1⊕R=(−25,15,∓45,±35,0,0),
5.17
g0 ⊕R=(−25−35,15−45,−45,35,0,0),
5.18
g−2,2⊕R=(−23+15,−25,+2.35,−2.45,0,0),
5.19
g−12,−1⊕R=(−23− 1.15,−25,−1.35,45,0,0),
5.19 2 2
g−1⊕3R=(−13,23,0,0,0,0),
3.4
g0 ⊕3R=(−23,13,0,0,0,0),
3.5
g ⊕g−1 =(−23,0,0,−46,56,0),
3.1 3.4
g ⊕g0 =(−23,0,0,−56,46,0),
3.1 3.5
g−1⊕g−1 =(−13,23,0,−46,56,0),
3.4 3.4
g−1⊕g0 =(−13,23,0,−56,46,0),
3.4 3.5
g0 ⊕g0 =(−23,13,0,−56,46,0),
3.5 3.5
where the parameters ε,p and r are real numbers.
To explain this notation,for example g0,0,r⊕R=(−25,15,−r.45,r.35,0,0) means
5.17
that there is a basis (α ,...,α ) of the dual of the Lie algebra g0,0,r⊕R such that
1 6 5.17
dα = −α ,dα = α ,dα = −rα ,dα = rα ,dα = 0,dα = 0, where by α
1 25 2 15 3 45 4 35 5 6 ij
we denote α ∧α .
i j
SIX DIMENSIONAL LIE ALGEBRAS 4
Proof. To construct the symplectic form we take the generic element ω ∈ kerd ⊂
2(g∗) and we impose it to be not degenerate, that is ω3 6=0.
VWiththis directcomputationwe canseethatthe six dimentionalsolvableunimod-
ularLie algebrasnotlistedbelowhavealwaysω3 =0foreveryω ∈kerd⊂ 2(g∗).
In Table 3 (Appendix) we list the symplectic structures admitted. V (cid:3)
Describingnilmanifoldsandsolvmanifoldswithsymplecticstructurebecameim-
portantafterthe workofThurston,[17]. Forthisreasonin[6],Guanstudiedprop-
erties about the steps of nilmanifolds, showing that if a nilmanifold G/Γ admits a
symplectic structure then G has to be at most two step as a solvable Lie group.
He also conjectured that the Lie group of a solvmanifold admitting a symplectic
structure is at most 3-step solvable.
We show by direct computation that this is true for all six dimensional uni-
modular solvable Lie algebra, also for those which do not admit any symplectic
structure.
Proposition 1. Every six dimensional unimodular, solvable, non-nilpotent Lie al-
gebra g is 2 or 3-step solvable, in particular
• if its nilradical has codimension 1, it is 3-step solvable unless it is almost
abelian, or g is one of the following Lie algebras:
ga,0 , g , g0,0 , g , g0 , g0,0,ε, g−1,0, g0,0,ε,
6.14 6.17 6.18 6.20 6.21 6.23 6.25 6.29
g0,0 , g0,−1, g , g0,0 , g0,0 , g0,0,0.
6.36 6.54 6.63 6.65 6.70 6.88
• if its nilradical has codimension greater then 1, it is 2-step solvable unless
g is one of the following Lie algebras:
g , g , g ⊕R, g ⊕R, g ⊕R, g ⊕R, g ⊕
6.129 6.135 5.19 5.20 5.23 5.25 5.26
R, g ⊕R, g ⊕R, g ⊕2R, g ⊕2R.
5.28 5.30 4.8 4.9
2. Betti numbers of 6-dimensional unimodular solvable
non-nilpotent Lie algebras
InthisSectionwecomputethesecondandthirdBettinumberofsixdimensional
solvable Lie algebras. The interest in determining solvable Lie algebras with the
propertythatb (g)=b (g)comesfromaclassofmanifoldsendowedwithaclosed3
2 3
form,calledStringgeometry,consideredin[9]. Stronggeometryisanimportantex-
ampleofconnectionbetweenmathematicsandphysics,inparticularmulti-moment
maps are used in string theory and one-dimensional quantum mechanics, [9].
LetM be amanifold,then(M,γ)isaStrong geometry ifγ is aclosed3-formon
M. Suppose there is a Lie group G that acts on M preserving γ, then we denote
by P the kernel of the map 2g→g induced by the Lie bracket of g.
g
A Multi-moment map is an eVquivariant map ν : M → Pg∗ such that dhν,pi = ipγ,
for any p∈P , (where i denotes the interior product) [8].
g p
We refer to [8] and [9] for details on strong geometry. In particular Madsen and
Swann [9] proved that if b (g)=b (g)=0, then there exists a multi-moment map
2 3
for the action of G on the manifold M. Because of this result they listed the Lie
algebraswithtrivialsecondandthirdBettinumbers,uptodimensionfive. Weadd
to their classification the Betti numbers of 6-dimensional solvable, non-nilpotent
unimodular Lie algebras.
SIX DIMENSIONAL LIE ALGEBRAS 5
Remark 1. Every Lie algebra g whose Lie group is solvable has b (g)>0, [1].
1
Next we list 6-dimensional unimodular, solvable, non-nilpotent Lie algebra g
togetherwiththeir first, secondandthird Bettinumber. The Bettinumbers ofthe
6-dimensionalLiealgebraswith5-dimensionalnilradicalwerealsocomputedbyM.
Freibert and F. F. Schulte-Hengesbach [4].
Looking at this list and comparing with Table 3 yields Theorem 1.
Table 1: Betti numbersof 6 dimensional unimodular, solvable, non-nilpotent Lie algebras 1
g b b b
1 2 3
g 1 0 if a6=−1, b6=−1, b6=−a, 0 if a6=−1, b6=−1, b6=−a,
6.1
c6=−a, c+b6=−1, c6=−b, c6=−a, c+b6=−1, c6=−b,
a+b6=−1, a+c6=−1 a+b6=−1, a+c6=−1
1 if a=−1, or if b=−a, 2 if a=−1, or if b=−a,
orifb=−c,orifa+b6=−1 orifb=−c,orifa+b6=−1
2 if b=−1, 4 if b=−1,
orifc=−aorifc=−1−a, orifc=−aorifc=−1−a,
or if a=−1 and b=1, or if a=−1 and b=1,
orifa=−1andb+c=−1, orifa=−1andb+c=−1,
or if b=c=−a, or if b=c=−a,
or if b=−c=±a, or if b=−c=±a,
or if b=−c=±(1+a), or if b=−c=±(1+a),
or if b=c=−1−a or if b=c=−1−a
3 if a=−1 and b=−c=±1 6 if a=−1 and b=−c=±1
2 2 2 2
or if a=b=c− 1 or if a=b=c− 1
2 2
4 if a=−b=c= 1 5 if a=−b=c= 1
2 2
g 1 if a6=0 0 if a6=0, c6=−1, e6=−c, 0 if a6=0, c6=−1, e6=−c,
6.2
c−e6=±1, c+e6=1, c−e6=±1, c+e6=1,
2 if a=0 1 if c=−1, 2 if a=0 or if c=−1
or if e=−c, or if e=c+1, or if e=−c, or if e=c+1,
or if e=±(1−c), or if e=±(1−c),
2 if a=0 or if c=−e=±1 4 if c=−e=±1
2 2
or if e=−1, or if e=−1,
or if c=−e=−1 or if c=−e=−1
g 1 if a6=−1 0 if a6=−1,1 0 if a6=−1,1
6.3 2 2
2 if a=−1 1 if a= 1 2 if a= 1
2 2
3 if a=−1 4 if a=−1
g 1 0 0
6.4
g 1 if a6=−1 0 if a6=−1,−1 0 if a6=−1,−1
6.6 2 2 2
2 if a=−1 1 if a=−1 2 if a=−1
2
2 if a=−1 2 if a=−1
2 2
1InTable1weimposeconditions whichbecomeateverystepmorerestrictive. Itistherefore
implicitthatthepreviousconditionsholdonlywhenthemorerestrictiveonesarenotsatisfied.
SIX DIMENSIONAL LIE ALGEBRAS 6
g b b b
1 2 3
g 1 0 0
6.7
g 1 0 if a+b6=0, a+c6=0, 0 if a+b6=0, a+c6=0,
6.8
b+c6=0, p6=0 b+c6=0, p6=0
1 if a+b=0 , 2 if a+b=0,
or if b+c=0, or if b+c=0,
or if p=0 or if p=0
2 if a=−b=c, 4 if a=−b=c ,
or if a+c=0 or if a+c=0
g 1 if b6=0 0 if bp6=0, a+b6=0 0 if bp6=0, a+b6=0
6.9
2 if b=0 1 if p=0 or if a+b=0 2 if bp=0 or if a+b=0
2 if b=0
g 1 if a6=0 0 if a6=0 0 if a6=0
6.10
2 if a=0 3 if a=0 4 if a=0
g 1 0 if pq6=0 0 if pq6=0
6.11
1 if pq=0 2 if pq=0
g 1 0 0
6.12
g 1 if bh6=0 0 if a6=−1, b6=−1, 0 if a6=−1, b6=−1,
6.13
a+b6=0, 2a+b6=0, a+b6=0, 2a+b6=0,
a+2b6=0, a+2b+16=0, a+2b6=0, a+2b+16=0,
b+2a+16=0 b+2a+16=0
2 if b=0 1 if a=−1 or if b=−1 2 if a=−1 or if b=−1
or if h=0 or if a+b=0 or if a+b=0
or if a+2b=0,−1 or if a+2b=0,−1
or if b+2a=0,−1 or if b+2a=0,−1
2 if a=−1 and b=2 4 if a=−1 and b=2
or if b=−1 and a=2 or if b=−1and a=2
or if a= 1 and b=−2 or if a= 1 and b=−2
3 3 3 3
or if a=−2 and b= 1 or if a=−2 and b= 1
3 3 3 3
or if a=b=−1,−1 or if a=b=−1,−1
3 3
3 if a= 1 and b=−1 4 if a= 1 and b=−1
2 2
or if a=−1 and b=0,1 or if a=−1 and b=0,1
2 2
or if a=−b=±1 6 if a=−b=±1
g 1 if a6=−1 0 if a6=−1,−2,−1,1,2 0 if a6=−1,−2,1,2
6.14 3 3 3 3 3 3 3 3
2 if a=−1 1 if a=−1,−2,−1,1,2 2 if a6=−1,−2,1,2
3 3 3 3 3 3 3 3
g 1 2 4
6.15
g 2 2 1
6.17
g 1 if a6=0 0 if a6=0,−1,−1,−2,−3 0 if a6=−1,−1,−2,−3
6.18 2 2
2 if a=0 1 if a=0,−1,−2,−3 2 if a=−1,−2,−3
2 2
2 if a=−1 4 if a=−1
g 1 0 0
6.19
g 2 1 0
6.20
g 1 if a6=0 0 if a6=0,−1,−1 0 if a6=0,−1,−1
6.21 3 3
2 if a=0 1 if a=−1,−1 2 if a=−1,−1
3 3
3 if a=0 4 if a=0
g 1 0 0
6.22
g 1 if a6=0 0 if a6=0 0 if a6=0
6.23
3 if a=0 5 if a=0 6 if a=0
g 1 if 0 if b6=0,−1,−1,1 0 if b6=0,−1,−1,1
6.25 2 2
b6=0,−1
2 if 1 if b=−1,1 2 if b=0,−1,−1,1
2 2
b=0,−1
2 if b=0,−1
SIX DIMENSIONAL LIE ALGEBRAS 7
g b b b
1 2 3
g 2 2 2
6.26
g 1 1 2
6.27
g 1 0 0
6.28
g 1 if b6=0 2 if b6=0 2 if b6=0
6.29
3 if b=0 5 if b=0 and ε6=0 6 if b=0 and ε6=0
6 if b=0 and ε=0 8 if b=0 and ε=0
g 1 ifa6=−h,−h 0 if a6=0,−h,−h 0 if a6=0
6.32 2 6 2 6
2 ifa=−h,−h 1 if a=0,−h,−h 2 if a=0
2 6 2 6
g 1 if a6=0 0 if a6=0 0 if a6=0
6.33
3 if a=0 3 if a=0 1 if a=0
g 1 if a6=0 0 if a6=0 0 if a6=0
6.34
3 if a=0 3 if a=0 1 if a=0
g 1 if ab6=0 0 if c6=0,a6=0,−2b, 0 if c6=0,a6=−2b,
6.35
b6=0,−2a b6=−2a
2 if a=0 1 ifa=0orifb=0orifc=0 2 if c=0 or if a=−2b
or b=0 or if a=−2b or if b=−2a or if b=−2a
g 1 if a6=0 0 if a6=0 0 if a6=0
6.36
2 if a=0 3 if a=0 4 if a=0
g 1 0 if a6=0 0 if a6=0
6.37
1 if a=0 2 if a=0
g 1 2 4
6.38
g 1 if h6=0 0 if h6=0,−1,−1,−2,−3 0 if h6=−1,−1,−2,−3
6.39 2 2
2 if h=0 1 if h=0,−1,−1,−2,−3 2 if h=−1,−1,−2,−3
2 2
g 1 0 0
6.40
g 1 1 2
6.41
g 1 0 0
6.42
g 1 0 0
6.44
g 2 1 0
6.47
g 1 if l6=−1, 0 if l6=0,−1,−1,−2,−3,−2 0 if l6=0,−1,−3,−2
6.54 2 2 2 3 2 3
−1,−2 1 if l=0,−1,−2,−3,−2 2 if l=0,−3,−2
2 2 3 2 3
2 if l=−1, 3 if l=−1 4 if l=−1
2
−1,−2
g 1 0 0
6.55
g 1 0 0
6.56
g 1 0 0
6.57
g 1 0 0
6.61
g 2 2 2
6.63
g 1 if l6=0 0 if l6=0 0 if l6=0
6.65
3 if l=0 5 if l=0 6 if l=0
g 1 if p6=0 0 if p6=0 0 if p6=0
6.70
2 if p=0 3 if p=0 4 if p=0
g 1 0 0
6.71
g 1 1 1
6.76
g 1 1 2
6.78
g 1 if l6=0 1 if l6=0 2 if l6=0
6.83
3 if l=0 5 if l=0 6 if l=0
g 2 2 2
6.84
g 1 if µ6=0 or 1 if µν 6=0 2 µν 6=0
6.88
ν 6=0 3 if µ6=0 and ν =0 6 if µ6=0 and ν =0
5 if µ=0 and or if µ=0 and ν 6=0 or if µ=0 and ν 6=0
ν =0 9 if µ=0 and ν =0 10if µ=0 and ν =0
SIX DIMENSIONAL LIE ALGEBRAS 8
g b b b
1 2 3
g 1 if sν 6=0 1 if sν 6=0 2 if sν 6=0
6.89
2 if s6=0 3 if s6=0 and ν =0 or if s6=0 and ν =0
or ν 6=0 or if s=0 and ν 6=0 or if s=0 and ν 6=0
5 if ν =0 9 if s=0 and ν =0 10if s=0 and ν =0
and s=0
g 1 if ν 6=0 1 if ν 6=0 2
6.90
3 if ν =0 3 if ν =0
g 1 1 2
6.91
g 1 if µν 6=0 3 if µν 6=0 4 if µν 6=0
6.92
2 if µ6=0 5 if µ6=0and ν =0 6 if µ6=0 and ν =0
or if ν 6=0 or if µ=0 and ν 6=0 or if µ=0 and ν 6=0
5 if ν =0 9 if µ=0and ν =0 10if µ=0 and ν =0
and µ=0
g∗ 1 3 6
6.92
g 1 if ν 6=0 1 if ν 6=0 2 if ν 6=0
6.93
3 if ν =0 3 if ν =0 2 if ν =0
g 1 1 2
6.94
g 2 1 if a6=−2 or b6=−1 0 if a6=−2 or b6=−1
6.101
2 if a=−2 and b=−1 1 if a=−2 and b=−1
g 2 1 0
6.102
g 2 1 0
6.105
g 2 1 0
6.107
g 2 1 if a6=0 or b6=−1 0 if a6=0 or b6=−1
6.113
3 if a=0 and b=−1 2 if a=0 and b=−1
g 2 2 if a6=±2 2 if a6=±2
6.114
3 if a=±2 3 if a=±2
g 2 1 0
6.115
g 2 1 0
6.116
g 2 1 if b6=±1 0 if b6=±1
6.118
3 if b=±1 4 if b=±1
g 2 2 2
6.120
g 2 2 2
6.121
g 2 1 0
6.129
g 2 1 0
6.135
n 1 1 2
6.83
n 1 1 1
6.84
n 1 if b6=0 1 if b6=0 2
6.96
3 if b=0 3 if b=0
g ⊕R 2 1 if r6=−1 0 if r6=−1
5.7
3 if r=−1 and q6=−1 4 if r=−1 and q6=−1
5 if r=−1 and q=−1 8 if r=−1 and q=−1
g ⊕R 3 5 6
5.8
g ⊕R 2 if p6=0 1 if p6=0,−1 0 if p6=0,−1
5.9
3 if p=0 3 if p=0,−1 2 if p=0
4 if p=−1
g ⊕R 2 1 0
5.11
g ⊕R 2 1 if q6=0 0 if q6=0
5.13
3 if q=0 4 if q=0
g ⊕R 2 5 6
5.14
SIX DIMENSIONAL LIE ALGEBRAS 9
g b b b
1 2 3
g ⊕R 2 3 4
5.15
g ⊕R 2 1 0
5.16
g ⊕R 2 1 if p6=0 and r6=±1 0 if p6=0 and r6=±1
5.17
3 if p=0 and r6=±1 4 if p=0 and r6=±1
or if p6=0 and r=±1 or if p6=0 and r=±1
5 if p=0 and r=±1 8 if p=0 and r=±1
g ⊕R 2 3 4
5.18
g ⊕R 2 if p6=0 1 if p6=0,−1,−2 0 if p6=0,−1,−2
5.19 2 2
3 if p=0 3 if p=0,−1,−2 2 if p=0
2
4 if p=−1,−2
2
g ⊕R 3 3 3
5.20
g ⊕R 2 1 0
5.23
g ⊕R 2 1 0
5.25
g ⊕R 3 3 2
5.26
g ⊕R 2 1 0
5.28
g ⊕R 2 1 0
5.30
g ⊕R 3 3 2
5.33
g ⊕R 3 3 1
5.35
g ⊕2R 3 3 2
4.2
g ⊕2R 3 3 2
4.5
g ⊕2R 3 3 2
4.6
g ⊕2R 3 3 2
4.8
g ⊕2R 3 3 2
4.9
g ⊕3R 4 7 8
3.4
g ⊕3R 4 7 8
3.5
g ⊕g 3 5 6
3.1 3.4
g ⊕g 3 5 6
3.1 3.5
g ⊕g 2 3 4
3.4 3.4
g ⊕g 2 3 4
3.4 3.5
g ⊕g 2 3 4
3.5 3.5
3. Hard Lefschetz property of 6 dimensional unimodular
non-nilpotent solvmanifolds
L.S. Tseng and S.T. Yau introduced some classes of finite dimensional coho-
mologies for symplectic manifolds [18]. These cohomology classes depend on the
symplectic formand are in generaldistinct from the de Rham cohomology,so that
they provide new symplectic invariants. As shown in [18] (cf. also Proposition 3
below), these new invariants actually agree with the de Rham cohomology if and
only if the Hard Lefschetz property holds.
In this Section we discuss these cohomological invariants, proving that they
can be computed using invariant forms, provided this is the case for the Rham
cohomology (see Theorem 3). This result will allow us to go through the list of
symplecticstructuresonsolvableLiealgebras(Table3),toseewhichsolvmanifolds
satisfy the Hard Lefschetz property (Theorem 4).
SIX DIMENSIONAL LIE ALGEBRAS 10
Let (M,ω) be a symplectic manifold of dimension 2n, one defines the Lefschetz
operator
L:Ωk(M)→Ωk+2(M)
η 7→η∧ω
its dual operator Λ : Ωk(M) → Ωk−2(M), and the symplectic star operator
∗ :Ωk(M)→Ω2n−k(M), such that for any γ,β ∈Ωk(M),
s
γ∧∗ β :=(ω−1)k(γ,β)dvol.
s
Remark 2. (see [18])
(1) ∗ ∗ =1.
s s
(2) Using coordinates (x ,..,x ) on M the above operatorsare defined in the
1 2n
following way:
1
Λ(η):= (ω−1)iji i η
2 ∂xi ∂xj
where i is the interior product,
1 ωn
γ∧∗sβ := k!(ω−1)i1j1(ω−1)i2j2...(ω−1)ikjkγi1i2...ikβj1j2...jk n!.
Using Λ one can construct two other differential operators:
d∧ :=(−1)k+1∗ d∗ =dΛ−Λd and dd∧
s s
with which one can define the following cohomologies
kerdΛ∩Ωk(M)
Hk (M):=
dΛ imdΛ∩Ωk(M)
ker(d+dΛ)∩Ωk(M)
Hk (M):=
d+dΛ imddΛ∩Ωk(M)
kerddΛ∩Ωk(M)
Hk (M):=
ddΛ imd∩Ωk(M)+imdΛ∩Ωk(M)
ker(d+dΛ)∩Ωk(M)
Hk (M):=Hk∩Hk =Hk ∩Hk =
d∩dΛ d dΛ d+dΛ ddΛ imd∩Ωk(M)+imdΛ∩Ωk(M)
0 0
where Ωk(M) is kerddΛ∩Ωk(M).
0
We refer to [18] for details and for the following propositions
Proposition2. (Tseng-Yau)Theoperator∗ givesanisomorphismbetweenHk(M)
s d
and H2n−k and between Hk (M) and H2n−k.
dΛ d+dΛ ddΛ
Proposition 3. (Tseng-Yau) On a compact symplectic manifold (M,ω) the fol-
lowing properties are equivalent:
• the Hard Lefschetz property holds.
• the canonical homomorphism Hk (M)→Hk(M) is an isomorphism for
d+dΛ d
all k.
• the canonical homomorphism Hk (M) →Hk (M) is an isomorphism
d∩dΛ d+dΛ
for all k.
WeareinterestedintheLiegroupsassociatedtothesixdimensionalunimodular
solvablenon-nilpotentLiealgebraswhichadmitalatticeandforwhichthedeRham
cohomologycanbecomputedbyinvariantforms,e.g.,theyarecompletelysolvable.
Indeed the following theorem holds: