Infinitesimal deformations of the model $\mathbb{Z}_3$-filiform Lie algebra

0.18 MB·English

by  Rosa Navarro

INFINITESIMAL DEFORMATIONS OF THE MODEL
Z -FILIFORM LIE ALGEBRA
3
3
1 R.M.NAVARRO
0
2
Abstract. Inthisworkitisconsideredthevectorspacecomposedbythein-
n finitesimaldeformationsofthemodelZ3-filiformLiealgebraLn,m,p. Byusing
a thesedeformationsalltheZ3-filiformLiealgebrascanbeobtained, hencethe
J importanceofthesedeformations. Theresultsobtainedinthisworktogether
7 tothoseobtainedin[11]and[12],leadstocomputethetotaldimensionofthe
1 mentioned spaceofdeformations.

]
T
2000 MSC: 17B30; 17B70; 17B75; 17B56
R
Key-Words: graded Lie algebras, cohomology, deformation, nilpotent, filiform.
.
h
t 1. Introduction
a
m
The concept of filiform Lie algebras was firstly introduced in [18] by Vergne.
[ This type of nilpotent Lie algebra has important properties; in particular, every

filiform Lie algebra canbe obtainedby a deformationof the model filiform algebra
1
v Ln. In the same way as filiform Lie algebras, all filiform Lie superalgebras can be
7 obtained by infinitesimal deformations of the model Lie superalgebra Ln,m [1], [4],
4 [8] and [9].
0
4
Continuing with the work of Vergne we have generalized the concept and the
.
1
propierties of the filiform Lie algebras into the theory of color Lie superalgebras.
0
Thus, filiform G-color Lie superalgebras and the model filiform G-color Lie super-
3
1 algebra were obtained in [10].
:
v
In the present paper the focus of interest are color Lie superalgebras with a
i
X Z -grading vector space, i.e. G = Z , due to its physical applications [3],[6],[7],
3 3
r [13], [16] and [17]. Due to the fact that the one admissible commutation factor
a for Z is exactly β(g,h) = 1 ∀g,h, Z -color Lie superalgebras are indeed Z -color
3 3 3
Lie algebras or Z -graded Lie algebras. Thus, we have studied the infinitesimal
3
deformationsof the model Z -colorLie superalgebra,i.e. the model Z -filiform Lie
3 3
algebra Ln,m,p. By means of these deformations all Z -filiform Lie algebras can be
3
obtained, hence the importance of these deformations.
In [11] and [12], the authors decomposed the space of these infinitesimal defor-
mations, noted by Z2(L;L), into six subspaces of deformations:
Z2(L;L)∩Hom(L ∧L ,L )⊕Z2(L;L)∩Hom(L ∧L ,L )⊕
0 0 0 0 1 1
Z2(L;L)∩Hom(L ∧L ,L )⊕Z2(L;L)∩Hom(L ∧L ,L )⊕
0 2 2 1 1 2
Z2(L;L)∩Hom(L ∧L ,L )⊕Z2(L;L)∩Hom(L ∧L ,L )
1 2 0 2 2 1
=A⊕B⊕C⊕D⊕E⊕F
1
2 R.M.NAVARRO
In the present paper it is given a method that will allow to determine the di-
mension of the subspaces A, B and C, giving explicitly the total dimension of all
of them (Theorems 1, 2 and 3). This result together to those obtained in [11] and
[12], leads to obtain the total dimension of the infinitesimal deformations of the
model Z -filiform Lie algebra Ln,m,p (Main Theorem).
3
WedoassumethatthereaderisfamiliarwiththestandardtheoryofLiealgebras.
All the vector spaces that appear in this paper (and thus, all the algebras) are
assumed to be F-vector spaces (F=C or R) with finite dimension.
2. Preliminaries
The vector space V is said to be Z −graded if it admits a decomposition in
n
direct sum, V = V0⊕V1⊕···Vn−1. An element X of V is called homogeneous of
degree γ (deg(X)=d(X)=γ), γ ∈Z , if it is an element of V .
n γ
Let V =V0⊕V1⊕···Vn−1 and W =W0⊕W1⊕···Wn−1 be two gradedvector
spaces. A linear mapping f : V −→ W is said to be homogeneous of degree γ
(deg(f)=d(f)=γ), γ ∈Z , iff(V )⊂W forall α∈Z . The mapping
n α α+γ(mod n) n
f is calleda homomorphismofthe Z −gradedvectorspace V into the Z −graded
n n
vector space W if f is homogeneous of degree 0. Now it is evident how we define
an isomorphism or an automorphism of Z −graded vector spaces.
n
A superalgebrag is just a Z −gradedalgebra g=g ⊕g . That is, if we denote
2 0 1
by [ , ] the bracket product of g, we have [g ,g ]⊂g for all α,β ∈Z .
α β α+β(mod2) 2
Definition 2.1. [14]Let g=g ⊕g be a superalgebrawhose multiplicationis de-
0 1
notedbythebracketproduct[,]. WecallgaLiesuperalgebraifthemultiplication
satisfies the following identities:
1. [X,Y]=−(−1)α·β[Y,X], ∀X ∈g ,∀Y ∈g .
α β
2. (−1)γ·α[X,[Y,Z]]+(−1)α·β[Y,[Z,X]]+(−1)β·γ[Z,[X,Y]]=0
for all X ∈g ,Y ∈g ,Z ∈g with α,β,γ ∈Z .
α β γ 2
Identity2iscalledthegradedJacobiidentityanditwillbedenotedbyJ (X,Y,Z).
g
We observe that if g = g ⊕g is a Lie superalgebra, we have that g is a Lie
0 1 0
algebra and g has the structure of a g −module.
1 0
Color Lie (super)algebras can be seen as a direct generalization of Lie (su-
per)algebras. Indeed, the latter are defined through antisymmetric (commutator)
or symmetric (anticommutator) products, although for the former the product is
neither symmetric nor antisymmetric and is defined by means of a commutation
factor. This commutation factor is equal to ± 1 for (super)Lie algebras and more
generalfor arbitrarycolor Lie (super)algebras. As happened for Lie superalgebras,
the basic tool to define color Lie (super)algebras is a grading determined by an
abelian group.
Definition 2.2. Let G be an abelian group . A commutation factor β is a map
β : G×G−→F\{0}, (F=C or R), satisfying the following constraints:
(1) β(g,h)β(h,g)=1 for all g,h∈G
(2) β(g,h+k)=β(g,h)β(g,k) for all g,h,k∈G
INFINITESIMAL DEFORMATIONS OF THE MODEL Z3-FILIFORM LIE ALGEBRA 3
(3) β(g+h,k)=β(g,k)β(h,k) for all g,h,k∈G
The definition above implies, in particular, the following relations:
β(0,g)=β(g,0)=1, β(g,h)=β(−h,g), β(g,g)=±1 ∀g,h∈G
where 0 denotes the identity element of G. In particular, fixing g one element of
G, the induced mapping β :G−→F\{0} defines a homomorphism of groups.
g
Definition 2.3. Let G be an abelian group and β a commutation factor. The
(complex or real) G−graded algebra
L= Lg
g∈G
M
with bracket product [ , ], is called a (G,β)-color Lie superalgebra if for any X ∈
L , Y ∈L , and Z ∈L we have
g h
(1) [X,Y]=−β(g,h)[Y,X] (anticommutative identity)
(2) [[X,Y],Z]=[X,[Y,Z]]−β(g,h)[Y,[X,Z]] (Jacobi identity)
Corollary 2.3.1. Let L = Lg be a (G,β)-color Lie superalgebra. Then we
g∈G
have
L
(1) L is a (complex or real) Lie algebra where 0 denotes the identity element
0
of G.
(2) For all g ∈ G\{0}, L is a representation of L . If X ∈ L and Y ∈ L ,
g 0 0 g
then [X,Y] denotes the action of X on Y.
Examples. For the particular case G = {0}, L = L reduces to a Lie algebra.
0
If G = Z = {0,1} and β(1,1) = −1 we have ordinary Lie superalgebras, i.e. a
2
Lie superalgebra is a (Z ,β)-color Lie superalgebra where β(i,j) = (−1)ij for all
2
i,j ∈Z .
2
Definition 2.4. A representation of a (G,β)-color Lie superalgebra is a mapping
ρ:L−→End(V), where V = V is a graded vector space such that
g∈G g
[ρ(X),ρ(Y)]L=ρ(X)ρ(Y)−β(g,h)ρ(Y)ρ(X)
for all X ∈L , Y ∈L
g h
We observe that for all g,h ∈ G we have ρ(L )V ⊆ V , which implies that
g h g+h
any V has the structure of a L -module. In particular considering the adjoint
g 0
representationad we have that every L has the structure of a L -module.
L g 0
Two (G,β)-color Lie superalgebras L and M are called isomorphic if there is
a linear isomorphism ϕ : L −→ M such that ϕ(L ) = M for any g ∈ G and also
g g
ϕ([x,y])=[ϕ(x),ϕ(y)] for any x,y ∈L.
LetL= Lg bea(G,β)-colorLiesuperalgebra. Thedescending central
g∈G
sequence of L is defined by
L
C0(L)=L, Ck+1(L)=[Ck(L),L] ∀k ≥0
If Ck(L) = {0} for some k, the (G,β)-color Lie superalgebra is called nilpotent.
The smallest integer k such as Ck(L)={0} is called the nilindex of L.
4 R.M.NAVARRO
Also, we are going to define some new descending sequences of ideals, see
[10]. Let L = Lg be a (G,β)-color Lie superalgebra. Then, we define the
g∈G
new descending sequences of ideals Ck(L ) (where 0 denotes the identity element
L 0
of G) and Ck(L ) with g ∈G\{0}, as follows:
g
C0(L )=L , Ck+1(L )=[L ,Ck(L )], k ≥0
0 0 0 0 0
and
C0(L )=L , Ck+1(L )=[L ,Ck(L )], k ≥0, g ∈G\{0}
g g g 0 g
Using the descending sequences of ideals defined above we give an invariant of
color Lie superalgebras called color-nilindex. We are going to particularize this
definition for G=Z .
3
Definition 2.5. [11] If L = L ⊕L ⊕L is a nilpotent (Z ,β)-color Lie superal-
0 1 2 3
gebra, then L has color-nilindex (p ,p ,p ), if the following conditions hold:
0 1 2
(Cp0−1(L ))(Cp1−1(L ))(Cp2−1(L ))6=0
0 1 2
and
Cp0(L )=Cp1(L )=Cp2(L )=0
0 1 2
Definition 2.6. [10] Let L = Lg be a (G,β)-color Lie superalgebra. L is
g∈G g
called a L -filiform module if there exists a decreasing subsequence of vectorial
0
L
subspaces in its underlying vectorial space V, V = V ⊃ ··· ⊃ V ⊃ V , with
m 1 0
dimensions m,m−1,...0, respectively, m>0, and such that [L ,V ]=V .
0 i+1 i
Remark 2.7. ThedefinitionoffiliformmoduleisalsovalidforG-gradedLiealgebras.
Definition 2.8. [10] Let L = Lg be a (G,β)-color Lie superalgebra. Then
g∈G
L is a filiform color Lie superalgebra if the following conditions hold:
L
(1) L is a filiform Lie algebra where 0 denotes the identity element of G.
0
(2) L has structure of L -filiform module, for all g ∈G\{0}
g 0
Definition 2.9. Let L = Lg be a G-graded Lie algebra. Then L is a G-
g∈G
filiform Lie algebra if the following conditions hold:
L
(1) L is a filiform Lie algebra where 0 denotes the identity element of G.
0
(2) L has structure of L -filiform module, for all g ∈G\{0}
g 0
It is not difficult to see that for G = Z , there is only one possibility for the
3
commutation factor β, i. e.
β(g,h)=1 ∀ g, h ∈Z ={0,1,2}
3
From now on we will consider this commutation factor and we will write “Z -
3
color”instead of “(Z ,β)-color”. We will note by Ln,m,p the variety of all Z -color
3 3
Lie superalgebras L = L ⊕L ⊕L with dim(L ) = n+1, dim(L ) = m and
0 1 2 0 1
dim(L )=p. Nn,m,p will be the variety ofall nilpotent Z -colorLie superalgebras
2 3
andFn,m,p isthe subsetofNn,m,p composedofallfiliformcolorLiesuperalgebras.
Remark 2.10. If G = Z then β(g,h) = 1 ∀ g, h. Thus, Z -color Lie super-
3 3
algebras are effectively Z -graded Lie algebras and filiform Z -color Lie
3 3
superalgebras are Z -filiform Lie algebras.
3
INFINITESIMAL DEFORMATIONS OF THE MODEL Z3-FILIFORM LIE ALGEBRA 5
In the particular case ofG=Z the theoremof adaptedbasis restas follows for
3
L=L ⊕L ⊕L ∈Fn,m,p:
0 1 2
[X ,X ]=X , 1≤i≤n−1,
0 i i+1
[X ,X ]=0,
 0 n
 [[XX00,,YYmj]]==Y0j+1, 1≤j ≤m−1,
[X ,Z ]=Z , 1≤k ≤p−1,
0 k k+1
with {X0,X1,..., Xn}a[bXa0s,isZpo]f=L00,.{Y1,...,Ym} a basis of L1 and {Z1,...,Zp}
abasis ofL . The model Z -filiformLie algebra,Ln,m,p, is the simplestZ -filiform
2 3 3
Lie algebra and it is defined in an adapted basis {X ,X ,...,X , Y ,...,Y ,
0 1 n 1 m
Z ,...,Z } by the following non-null bracket products
1 p
[X ,X ]=X , 1≤i≤n−1
0 i i+1
Ln,m,p = [X ,Y ]=Y , 1≤j ≤m−1
 0 j j+1
 [X0,Zk]=Zk+1 1≤k ≤p−1
3. cocycles and infinitesimal deformations
Recall that a module V = V ⊕V ⊕V of the Z -color Lie superalgebra L is a
0 1 2 3
bilinear map of degree 0, L×V →V satisfying
∀ X ∈L , Y ∈L v ∈V : X(Yv)−Y(Xv)=[X,Y]v
g h
color Lie superalgebra cohomology is defined in the following well-known way (see
e.g. [15]): inparticular,thesuperspaceofq-dimensional cocyclesoftheZ -colorLie
3
superalgebraL=L ⊕L ⊕L with coefficients in the L-module V =V ⊕V ⊕V
0 1 2 0 1 2
will be given by
Cq(L;V)= Hom(∧q0L ⊗∧q1L ⊗∧q2L ,V)
0 1 2
q0+qM1+q2=q
This space is graded by Cq(L;V)=Cq(L;V)⊕Cq(L;V)⊕Cq(L;V) with
0 1 2
Cq(L;V)= Hom(∧q0L ⊗∧q1L ⊗∧q2L ,V )
p 0 1 2 r
q0+q1M+q2=q
q1+2q2+p≡rmod3
The coboundary operator δq : Cq(L;V) −→ Cq+1(L;V), with δq+1 ◦δq = 0 is
defined in general, with L an arbitrary (G,β)-color Lie superalgebra and V an
L-module, by the following formula for q ≥1
(δqg) A ,A ,...,A =
0 1 q
q
(cid:0) (cid:1) (−1)rβ(γ+α0+···+αr−1,αr)Ar ·g A0,...,Aˆr,...,Aq
r=0
+ (−1)sβ(αr+1+X···+αs−1,αs)g A0,...,Ar−1,[Ar,As],A(cid:0)r+1,...,Aˆs,...Aq (cid:1),
r<s
X (cid:0) (cid:1)
where g ∈ Cq(L;V) of degree γ, and A ,A ,...,A ∈ L are homogeneous with
0 1 q
degreesα ,α ,...,α respectively. Thesignˆindicatesthattheelementbelowmust
0 1 q
6 R.M.NAVARRO
be omitted and empty sums (like α0+···+αr−1 for r =0 and αr+1+···+αs−1
for s=r+1) are set equal to zero. In particular, for q =2 we obtain
(δ2g) A ,A ,A = β(γ,α )A ·g A ,A −β(γ+α ,α )A ·g A ,A +
0 1 2 0 0 1 2 0 1 1 0 2
β(γ+α +α ,α )A ·g A ,A
0 1 2 2 0 1
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
−g([A ,A ],A )+β(α ,α )g([A ,A ],A )+g(A ,[A ,A ]).
0 1 2 1 2 0 2 1 0 1 2
(cid:0) (cid:1)
Let Zq(L;V) denote the kernelofδq and letBq(L;V)denote the imageof δq−1,
then we have that Bq(L;V) ⊂ Zq(L;V). The elements of Zq(L;V) are called q-
cocycles, the elements of Bq(L;V) are the q-coboundaries. Thus, we can constuct
the so-called cohomology groups
Hq(L;V)=Zq(L;V)/Bq(L;V)
Hq(L;V)=Zq(L;V) Bq(L;V),if G=Z then p=0,1,2
p p p 3
Two elements of Zq(L;V) are(cid:14)said to be cohomologous if their residue classes
modulo Bq(L;V) coincide, i.e., if their difference lies in Bq(L;V).
We will focus our study in the 2-cocycles Z2(Ln,m,p;Ln,m,p) with Ln,m,p the
0
model filiform Z -color Lie superalgebra. Thus G = Z and the only admissible
3 3
commutationfactorisexactlyβ(g,h)=1. Underalltheserestrictionsthecondition
that have to verify ψ ∈C2(Ln,m,p;Ln,m,p) to be a 2-cocycle rests
0
(δ2ψ) A ,A ,A = [A ,ψ A ,A ]−[A ,ψ A ,A ]+
0 1 2 0 1 2 1 0 2
[A ,ψ A ,A ]−ψ([A ,A ],A )+
2 0 1 0 1 2
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
ψ([A ,A ],A )+ψ(A ,[A ,A ])=0
0 2 1 0 1 2
(cid:0) (cid:1)
for all A ,A ,A ∈Ln,m,p. We observe that Ln,m,p has the structure of a Ln,m,p-
0 1 2
module via the adjoint representation.
We consider an homogeneous basis of Ln,m,p = L ⊕L ⊕L , in particular an
0 1 2
adapted basis {X ,X ,..., X ,Y ,...,Y ,Z ,...,Z } with {X ,X ,..., X } a
0 1 n 1 m 1 p 0 1 n
basis of L , {Y ,...,Y } a basis of L and {Z ,...,Z } a basis of L .
0 1 m 1 1 p 2
Under these conditions we have the following lemma.
Lemma 3.1. [11], [12] Let ψ be such that ψ ∈ C2(Ln,m,p;Ln,m,p), then ψ is a 2-
0
cocycle, ψ ∈Z2(Ln,m,p;Ln,m,p),iffthe10conditionsbelowholdforallX ,X ,X ∈
0 i j k
L , Y ,Y ,Y ∈L and Z ,Z ,Z ∈L
0 i j k 1 i j k 2
(1) [X ,ψ(X ,X )]−[X ,ψ(X ,X )]+[X ,ψ(X ,X )]−ψ([X ,X ],X )+
i j k j i k k i j i j k
ψ([X ,X ],X )+ψ(X ,[X ,X ])=0
i k j i j k
(2) [X ,ψ(X ,Y )]−[X ,ψ(X ,Y )]+[Y ,ψ(X ,X )]−ψ([X ,X ],Y )+
i j k j i k k i j i j k
ψ([X ,Y ],X )+ψ(X ,[X ,Y ])=0
i k j i j k
(3) [X ,ψ(X ,Z )]−[X ,ψ(X ,Z )]+[Z ,ψ(X ,X )]−ψ([X ,X ],Z )+
i j k j i k k i j i j k
ψ([X ,Z ],X )+ψ(X ,[X ,Z ])=0
i k j i j k
(4) [X ,ψ(Y ,Y )]−[Y ,ψ(X ,Y )]+[Y ,ψ(X ,Y )]−ψ([X ,Y ],Y )+
i j k j i k k i j i j k
ψ([X ,Y ],Y )+ψ(X ,[Y ,Y ])=0
i k j i j k
(5) [X ,ψ(Y ,Z )]−[Y ,ψ(X ,Z )]+[Z ,ψ(X ,Y )]−ψ([X ,Y ],Z )+
i j k j i k k i j i j k
ψ([X ,Z ],Y )+ψ(X ,[Y ,Z ])=0
i k j i j k
(6) [X ,ψ(Z ,Z )]−[Z ,ψ(X ,Z )]+[Z ,ψ(X ,Z )]−ψ([X ,Z ],Z )+
i j k j i k k i j i j k
ψ([X ,Z ],Z )+ψ(X ,[Z ,Z ])=0
i k j i j k
INFINITESIMAL DEFORMATIONS OF THE MODEL Z3-FILIFORM LIE ALGEBRA 7
(7) [Y ,ψ(Y ,Y )]−[Y ,ψ(Y ,Y )]+[Y ,ψ(Y ,Y )]−ψ([Y ,Y ],Y )+
i j k j i k k i j i j k
ψ([Y ,Y ],Y )+ψ(Y ,[Y ,Y ])=0
i k j i j k
(8) [Y ,ψ(Y ,Z )]−[Y ,ψ(Y ,Z )]+[Z ,ψ(Y ,Y )]−ψ([Y ,Y ],Z )+
i j k j i k k i j i j k
ψ([Y ,Z ],Y )+ψ(Y ,[Y ,Z ])=0
i k j i j k
(9) [Y ,ψ(Z ,Z )]−[Z ,ψ(Y ,Z )]+[Z ,ψ(Y ,Z )]−ψ([Y ,Z ],Z )+
i j k j i k k i j i j k
ψ([Y ,Z ],Z )+ψ(Y ,[Z ,Z ])=0
i k j i j k
(10) [Z ,ψ(Z ,Z )]−[Z ,ψ(Z ,Z )]+[Z ,ψ(Z ,Z )]−ψ([Z ,Z ],Z )+
i j k j i k k i j i j k
ψ([Z ,Z ],Z )+ψ(Z ,[Z ,Z ])=0
i k j i j k
Proposition 3.2. [11] ψ is an infinitesimal deformation of Ln,m,p iff ψ is a 2-
cocycle of degree 0, ψ ∈Z2(Ln,m,p;Ln,m,p).
0
Theorem 3.2.1. [10] (1) Any filiform (G,β)-color Lie superalgebra law µ is iso-
morphic to µ +ϕ where µ is the law of the model filiform (G,β)-color Lie super-
0 0
algebra and ϕ is an infinitesimal deformation of µ verifying that ϕ(X ,X)=0 for
0 0
all X ∈L, with X the characteristic vector of model one.
0
(2) Conversely, if ϕ is an infinitesimal deformation of a model filiform (G,β)-
color Lie superalgebra law µ with ϕ(X ,X)=0 for all X ∈L, then the law µ +ϕ
0 0 0
is a filiform (G,β)-color Lie superalgebra law iff ϕ◦ϕ=0.
Thus, any Z -filiform Lie algebra (filiform Z -color Lie superalgebra) will be a
3 3
linear deformation of the model Z -filiform Lie algebra (the model Z -color Lie
3 3
superalgebra), i.e. Ln,m,p is the model Z -filiform Lie algebra an another arbi-
3
trary Z -filiform Lie algebra will be equal to Ln,m,p+ϕ, with ϕ an infinitesimal
3
deformation of Ln,m,p. Hence the importance of these deformations. So, in or-
der to determine all the Z -filiform Lie algebras it is only necessary to compute
3
the infinitesimal deformations or so called 2-cocycles of degree 0, that vanish on
the characteristic vector X . Thanks to the following lemma these infinitesimal
0
deformations will can be decomposed into 6 subspaces.
Lemma3.3. [11],[12]LetZ2(L;L)bethe2-cocyclesZ2(Ln,m,p;Ln,m,p)thatvanish
0
on the characteristic vector X . Then, Z2(L;L) can be divided into six subspaces,
0
i.e. if Ln,m,p =L=L ⊕L ⊕L we will have that
0 1 2
Z2(L;L)= Z2(L;L)∩Hom(L ∧L ,L )⊕Z2(L;L)∩Hom(L ∧L ,L )⊕
0 0 0 0 1 1
Z2(L;L)∩Hom(L ∧L ,L )⊕Z2(L;L)∩Hom(L ∧L ,L )⊕
0 2 2 1 1 2
Z2(L;L)∩Hom(L ∧L ,L )⊕Z2(L;L)∩Hom(L ∧L ,L )
1 2 0 2 2 1
= A⊕B⊕C⊕D⊕E⊕F
In order to obtain the dimension of A, B and C we are going to adapt the
sl (C)-module method that we have already used for Lie superalgebras [1], [4], [8]
2
and for color Lie superalgebras [11], [12]. Next, we will do it explicitly for A =
Z2(L;L)∩Hom(L ∧L ,L ).
0 0 0
4. Dimension of A=Z2(L;L)∩Hom(L ∧L ,L )
0 0 0
In general, any cocycle a ∈ Z2(L;L) ∩ Hom(L ∧ L ,L ) will be any skew-
0 0 0
symmetric bilinear map from L ∧L to L such that:
0 0 0
(1) [X ,a(X ,X )]−[X ,a(X ,X )]+[X ,a(X ,X )]−a([X ,X ],X )+
i j k j i k k i j i j k
a([X ,X ],X )+a(X ,[X ,X ])=0 ∀ X ,X ,X ∈L
i k j i j k i j k 0
8 R.M.NAVARRO
with a(X ,X) = 0 ∀X ∈ L. As X ∈/ Im a and taking into account the bracket
0 0
products of L then the equation (1) can be rewritten as follows
(4.1) [X ,a(X ,X )]−a([X ,X ],X )−a(X ,[X ,X ])=0, 1≤j <k ≤n
0 j k 0 j k j 0 k
In order to obtain the dimension of the space of cocycles for A we apply an
adaptation of the sl(2,C)-module Method that we used in [11].
Recallthefollowingwell-knownfactsabouttheLiealgebrasl(2,C)anditsfinite-
dimensional modules, see e.g. [2], [5]:
sl(2,C)=<X−,H,X+ > with the following commutation relations:
[X+,X−]=H
[H,X ]=2X ,
 + +
 [H,X−]=−2X−.
LetV bean-dimensionalsl(2,C)-module,V =<e1,...,en >. Then,uptoisomor-
phism there exists a unique structure of an irreducible sl(2,C)-module in V given
in a basis {e ,...,e } as follows [2]:
1 n
X ·e =e , 1≤i≤n−1,
+ i i+1
X ·e =0,
 + n
 H ·ei =(−n+2i−1)ei, 1≤i≤n.
It is easy to seethat en is the maximal vector of V and its weight, called the
highest weight of V, is equal to n−1.
Let W ,W ,...,W be sl(2,C)-modules, then the space Hom(⊗k W ,W ) is a
0 1 k i=1 i 0
sl(2,C)-module in the following natural manner:
k
(ξ·ϕ)(x ,...,x )=ξ·ϕ(x ,...,x )− ϕ(x ,...,ξ·x ,x ,...,x )
1 k 1 k 1 i i+1 n
i=1
X
with ξ ∈ sl(2,C) and ϕ ∈ Hom(⊗k W ,W ). In particular, if k = 2 and W =
i=1 i 0 0
W =W =V , then
1 2 0
(ξ·ϕ)(x ,x )=ξ·ϕ(x ,x )−ϕ(ξ·x ,x )−ϕ(x ,ξ·x ).
1 2 1 2 1 2 1 2
An element ϕ∈Hom(V ⊗V ,V ) is said to be invariant if X ·ϕ=0, that is
0 0 0 +
(4.2) X ·ϕ(x ,x )−ϕ(X ·x ,x )−ϕ(x ,X ·x )=0, ∀x ,x ∈V.
+ 1 2 + 1 2 1 + 2 1 2
Note thatϕ∈Hom(V ⊗V ,V )isinvariantifandonlyifϕisamaximalvector.
0 0 0
We are going to consider the structure of irreducible sl(2,C)-module in V =<
0
X ,...,X >=L /CX , thus in particular:
1 n 0 0
X ·X =X , 1≤i≤n−1,
+ i i+1
( X+·Xn =0,
Next, we identify the multiplicationofX andX inthe sl(2,C)-module V =<
+ i 0
X ,...,X >, with the bracket [X ,X ] in L and thanks to these identifications,
1 n 0 i 0
the expressions (4.1) and (4.2) are equivalent. Thus we have the following result:
INFINITESIMAL DEFORMATIONS OF THE MODEL Z3-FILIFORM LIE ALGEBRA 9
Proposition 4.1. Any skew-symmetric bilinear map ϕ, ϕ:V ∧V −→V will be
0 0 0
an element of the space of cocycles A if and only if ϕ is a maximal vector of the
sl(2,C)-module Hom(V ∧V ,V ), with V =hX ,...,X i.
0 0 0 0 1 n
Corollary 4.1.1. As each irreducible sl(2,C)-module has (up to nonzero scalar
multiples) a unique maximal vector, then the dimension of the space of cocycles A
is equal to the number of summands of any decomposition of Hom(V ∧V ,V ) into
0 0 0
the direct sum of irreducible sl(2,C)-modules.
We use the fact that each irreducible module contains either a unique (up to
scalarmultiples)vectorofweight0(incasethedimensionoftheirreduciblemodule
isodd)oraunique(uptoscalarmultiples)vectorofweight1(incasethedimension
of the irreducible module is even). We therefore have
Corollary 4.1.2. The dimension of the space of cocycles A is equal to the dimen-
sion of the subspace of Hom (V ∧V ,V ) spanned by the vectors of weight 0 or
0 0 0
1.
At this point, we aregoingto apply the sl(2,C)-module method aforementioned
in order to obtain the dimension of the space of cocycles A.
We consider a natural basis B of Hom(V ∧V ,V ) consisting of the following
0 0 0
maps:
X if (i,j)=(k,l)
ϕs (X ,X )= s
i,j k l 0 in all other cases
(cid:26)
where 1≤i,j,k,l,s≤n, with i6=j and ϕs =−ϕs .
i,j j,i
Thanks to Corollary 5.1.2 it will be enough to find the basis vectors ϕs with
i,j
weight 0 or 1. The weight of an element ϕs (with respect to H) is
i,j
λ(ϕs )=λ(X )−λ(X )−λ(X )=n+2(s−i−j)+1.
i,j s i j
In fact,
(H ·ϕs )(X ,X ) =H ·ϕs (X ,X )−ϕs (H ·X ,X )−ϕs (X ,H ·X )
i,j i j i,j i j i,j i j i,j i j
=H ·X −ϕs ((−n−1+2i)X ,X )−ϕs (X ,(−n−1+2j)X )
s i,j i j i,j i j
=(−n−1+2s)X −(−n−1+2i)X −(−n−1+2j)X
s s s
=[n+2(s−i−j)+1]X
s
We observethat if n is even then λ(ϕ) is odd, and if n is odd then λ(ϕ) is even.
So, if n is even it will be sufficient to find the elements ϕs with weight 1 and if n
i,j
is odd it will be sufficient to find those of them with weight 0.
We can consider the three sequences that correspondwith the weights of V =<
X1,X2,...,Xn−1,Xn > in order to find the elements with weight 0 or 1:
−n+1,−n+3,...,n−3,n−1;
−n+1,−n+3,...,n−3,n−1;
−n+1,−n+3,...,n−3,n−1.
andwehavetocountthenumberofallpossibilitiestoobtain1(ifniseven)or0(if
n is odd). Remember that λ(ϕs )=λ(X )−λ(X )−λ(X ), where λ(X ) belongs
i,j s i j s
10 R.M.NAVARRO
to the last sequence, and λ(X ), λ(X ) belong to the first and second sequences
i j
respectively. Forexample,ifnisodd,wehavetoobtain0,sowecanfixanelement
(a weight) of the last sequence and then count the possibilities to sum the same
quantity between the two first sequences. Taking into account the skew-symmetry
ofϕs ,thatisϕs =−ϕs andi6=j, andrepeatingthe abovereasoningforallthe
i,j i,j j,i
elements of the last sequence we obtain the following theorem:
Theorem 1. Let Z2(L;L)be the 2-cocyclesZ2(Ln,m,p;Ln,m,p)thatvanishonthe
0
characteristic vector X . Then, if A=Z2(L;L)∩Hom(L ∧L ,L ) we have that
0 0 0 0
n(3n−2)
if n is even
8
dim A=
 3n2−4n+1 +⌊n+1⌋ if n is odd
8 4
Proof. It is convenient todistinguish the following four cases where the reasoning
for each case is not hard:
(1). n≡0 (mod 4).
(2). n≡1 (mod 4).
(3). n≡2 (mod 4).
(4). n≡3 (mod 4).
(cid:3)
5. Dimension of B =Z2(L;L)∩Hom(L ∧L ,L )
0 1 1
In general, any cocycle b ∈ Z2(L;L) ∩ Hom(L ∧ L ,L ) will be any skew-
0 1 1
symmetric bilinear map from L ∧L to L such that:
0 1 1
(2) [X ,b(X ,Y )]−[X ,b(X ,Y )]−b([X ,X ],Y )+
i j k j i k i j k
b([X ,Y ],X )+b(X ,[X ,Y ])=0 ∀ X ,X ∈L , Y ∈L
i k j i j k i j 0 k 1
with b(X ,X)=0 ∀X ∈L. This condition reduces to
0
(5.1)
[X ,b(X ,Y )]−b([X ,X ],Y )−b(X ,[X ,Y ])=0, 1≤j ≤n, 1≤k ≤m
0 j k 0 j k j 0 k
In order to obtain the dimension of the space of cocycles B we apply an adap-
tation of the sl(2,C)-module Method that we have already used in the precedent
section.
RecallthatifW ,W ,...,W aresl(2,C)-modules,thenthespaceHom(⊗k W ,W )
0 1 k i=1 i 0
will be a sl(2,C)-module in the following natural manner:
k
(ξ·ϕ)(x ,...,x )=ξ·ϕ(x ,...,x )− ϕ(x ,...,ξ·x ,x ,...,x )
1 k 1 k 1 i i+1 n
i=1
X