Bockstein Closed Central Extensions of Elementary Abelian

33 Pages·2010·0.28 MB·English

Bockstein Closed Central Extensions of Elementary
Abelian 2-Groups I: Binding Operators
Jonathan Pakianathan and Ergu¨n Yalc¸ın ∗
May 2, 2006
Abstract
Let E be a central extension of the form 0 → V → G →W → 0 where V and
W are elementary abelian 2-groups. Associated to E there is a quadratic map Q:
W →V given by the 2-power map which uniquely determines the extension. This
2
quadratic map also determines the extension class q of the extension in H (W,V)
and an ideal I(q) in H2(G,Z/2) which is generated by the components of q. We
say E is Bockstein closed if I(q) is an ideal closed under the Bockstein operator.
We findadirectconditiononthe quadraticformQthatcharacterizeswhenthe
extensionisBocksteinclosed. Usingthischaracterization,weshowforexamplethat
gqluna(dFr2a)tigcivfeonrmbsyiQnd(Auc)e=d Afro+mAt2hyeiefuldndBaomckenstteailnqculoasderdateixctfeonrsmionQs.gln : gln(F2) →
It is well known that an extension is Bockstein closed if and only if it lifts to
anextension0→M →G→W →0 for someZ/4[W]-lattice M. In this situation,
one may write β(q) = Lq for a “binding matrix” L with entries in H1(W,Z/2).
e
We find a direct way to calculate the module structure of M in terms of L. Using
this, we study extensions where the lattice M is diagonalizable/triangulable and
find interesting equivalent conditions to these properties.
2000 Mathematics Subject Classification. Primary: 20J05; Secondary: 17B50,
15A63.
Keywords. Group Extensions,Quadratic Maps, Groupcohomology,Restricted Lie
Algebras.
1 Introduction
It is of course well known that in studying the cohomology of finite groups the coho-
mology of p-groups plays a fundamental part. Any p-group P has a central series with
elementary abelian factors (for example the central Frattini series) and so in princi-
ple, its cohomology can be studied by iteratively considering central extensions with
elementary abelian kernel.
The simplest nontrivial situation is given by p-groups P that fit in a central short
exact sequence
0 → V → P → W → 0
where V and W are elementary abelian p-groups (finite F -vector spaces).
p
∗Partially supported bya grant from Turkish Academy of Sciences (TU¨BA-GEBI˙P/2005-16).
1
For p odd, such extensions are in bijective correspondence with a choice of p-power
map φ :W → V which is linear and an alternating bilinear commutator map
[·,·] :W ∧W → V.
For p = 2, such extensions are in bijective correspondence with a quadratic 2-power
map Q : W → V which has associated bilinear map given by the commutator. Since
the power map is not linear but quadratic in this case, this adds considerable difficulty
in this situation.
In this paper we study various algebraic properties of these extensions in the case
p = 2. Westartwithareviewoftheabovementionedfactsandanexplicitdescriptionof
how torecover theextension class inH2(W,V)oftheextension fromthequadraticmap
Q. The components of the extension class form a quadratic ideal I(Q) ⊆ H∗(W,F ).
2
In order to compute the cohomology of the group G given by the extension, an
importantconditionthatisoftenconsiderediswhethertheidealI(Q)isclosedunderthe
Bockstein (and hence under the Steenrod algebra). Our first theorem gives a necessary
and sufficient condition for the ideal I(Q) to be Bockstein closed based on conditions
on the associated quadratic form Q of the extension:
Theorem 1.1. Let Q : W → V be a quadratic map, and let B be the bilinear map
associated to Q. Then, Q is Bockstein closed if and only if there exists a bilinear map
P :V ×W → V which satisfies the identity
P(Q(x),y) = B(x,y)+P(B(x,y),x)
for all x,y ∈ W.
While this condition might seem mysterious at first, the case of P = B happens
when Q is obtained from the 2-power map of a 2-restricted Lie algebra (see Section 9).
A fundamental example of a quadratic map is Q : gl (F ) → gl (F ) given by
gln n 2 n 2
Q (A)= A+A2.
gl
n
Using the criterion above, it follows that any quadratic map induced from Q (by
gl
n
restriction of domain and codomain) has an associated extension which is Bockstein
closed.
Let G be the kernel of the mod 2 reduction map Gl (Z/8) → GL (Z/2). It is easy
n n
to see that G fits into a central short exact sequence
0 → gl (F ) → G → gl (F )→ 0
n 2 n 2
withassociatedquadraticformQ . Itfollowsthatthisextensionanditsrestrictionsto
gl
n
suitablesubspacessuchassl (F )oru (F )areBockstein closed. (Thisisanimportant
n 2 n 2
ingredient in calculating the cohomology of these groups as rings and as modules over
the Steenrod Algebra. See for example [1] and [4].)
In the case that E is a Bockstein closed extension, there is a cohomology class L in
H1(W,End(V)) such that β(q) = Lq holds. Here the multiplication Lq is given by the
composite
H1(W,End(V))⊗H2(W,V)−∪→H3(W,End(V)⊗V)→ H3(W,V)
2
where the second map is induced by the evaluation map ev : End(V)⊗V → V. In this
case we call the element L a binding operator. Note that if we choose a basis for V,
then we can write L as a matrix with entries in H1(W,F ) and express the equation
2
β(q) = Lq as a matrix equation, where q in the equation denotes the column matrix
whose entries are the components of q.
It is well known that the central extension E : 0 → V → G → W → 0 has a
“uniform” lift to an extension 0 → M → G → W → 0, where M is a free Z/4-
module with M/2M = V and G/2M = G, if and only if the extension E is Bockstein
e
closed. (We will reprove this in this paper when proving more general results.) The
e
lifting module M has a W-action that is in general nontrivial and this yields a map
ρ :W → GL(M)whereGL(M)denotestheautomorphismgroupofM. Notethatρ
M M
sitsinthesubgroupGL(M;V,V)ofGL(M)formedbyautomorphismsofM thatinduce
trivial action on M/2M and2M. In thepaper we derive a certain explicit exponent-log
correspondencebetweenHom(W,GL(M;V,V))andHom(W,End(V))andconcludethe
following:
Theorem 1.2. If E(Q) lifts to an extension with kernel M, then β(q) =[log∗(ρ )]q.
M
Conversely, if β(q) = Lq for some L ∈H1(W,End(V)), then E(Q) lifts to an extension
with kernel M where M has the representation exp∗(L) ∈Hom(W,GL(M;V,V)).
This allows one to compute a binding operator L from the module structure of a
lifting module M and vice-versa. Note that in general the lifting module is not unique,
but there is a unique lifting module for every chosen binding operator. In the paper we
give examples of extensions which have more than one lifting module and more than
one binding operator (see Example 3.5).
We then go on in Section 5 to study the conditions for a Bockstein closed extension
to have a diagonalizable binding operator L (or equivalently a diagonalizable lifting
module M). We get the following theorem characterizing the diagonalizable situation:
Theorem 1.3. Let Q : W → V be a Bockstein closed quadratic map, and let E : 0 →
V → G → W → 0 be the central extension associated to Q. Let q denote the extension
class for E. Then, the following are equivalent:
(i) Q is diagonalizable, i.e., there is a basis for V such that the components of q are
individually Bockstein closed.
(ii) There is a choice of basis of V, such that the components q ,...,q of q all decom-
1 n
pose as q = u v where u ,v are linear polynomials.
i i i i i
(iii) There exists a diagonalizable L ∈ Hom(W,End(V)) such that β(q) = Lq. This is
characterized exactly by the equation β(L)+L2 = 0.
(iv) E lifts to an extension with kernel M where M is a direct sum of one dimensional
Z/4[W]-lattices.
(v) E lifts to an extension with kernel M where M is a Z/8[W]-lattice.
(vi) E lifts to an extension with kernel M where M is a Z[W]-lattice, i.e., E has a
uniform integral lifting.
Some of the implications above are well known to experts on this subject. Themost
interesting implication in the above theorem is that if E has a uniform integral lifting,
then the associated quadratic map is diagonalizable (see Proposition 5.6). In other
words, the only extensions we get from integral extensions are the diagonalizable ones.
3
In Section 6, we consider triangulability of extensions and findequivalent conditions
to triangulability in terms of the binding operator L and lifting module M. We show
that if the extension is triangulable then it has a binding matrix L such that β(L)+L2
is nilpotent. Using this we give examples of extensions which are not triangulable.
In Section 7, we show that the components q ,...,q of q form a regular sequence
1 n
as long as we make the additional assumptions that dim(V) = dim(W) and that Q is
effective, i.e., that Q(w) = 0 is zero only when w = 0. This turns out to be another
important technical tool in calculating the cohomology of the groups given by these
extensions.
In Section 8, we consider Bockstein closed 2-power exact extensions (see Definition
7.9). We show thatfor aBockstein closed 2-power exact extension the bindingoperator
L, and hence lifting module M, is unique.
In Section 9, we show that any 2-restricted Lie algebra L gives a Bockstein closed
extensionoftheformE : 0 → L → G → L→ 0andfurthermorethatthereisabijection
between the 2-restricted Lie algebras over F and the Bockstein closed extensions with
2
P = B in Theorem 1.1.
In this paper we leave some questions about triangulability of Bockstein closed 2-
power exact extensions unanswered. These questions have equivalent versions in terms
of 2-restricted Lie algebras which may be easier to answer than the original versions.
The statements of these questions can be found in Sections 8 and 9.
2 Preliminaries and Definitions
Throughout this section, let E denote a central extension of the form
i π
0→ V −→G−→W → 0
where V and W are elementary abelian 2-groups. In this section, we will explain how
the standard theory of group extensions and 2-groups applies to E, then we will make
further definitions.
FirstwewillrecallhowthecohomologyclassassociatedtoE isdefined. Atransver-
sal is a function s : W → G such that π ◦s = id and s(0) = 1, where 0 ∈ W, 1 ∈ G
are the respective identities. Given a transversal s, we define f : W ×W → V by the
formula
f(x,y) =i−1 s(x+y)−1s(x)s(y) .
(cid:0) (cid:1)
We often identify V with its image under i and do not write i−1 in our formulas. It is
well-known that f satisfies the following normalized cocycle identities:
f(x,0) = f(0,x) = 0
f(x+y,z)+f(x,y)= f(y,z)+f(x,y+z)
Afunctionf : W×W → V satisfyingtheseidentitiesiscalleda(normalized)factorset,
and it is known that normalized factor sets correspond to 2-cocycles in the normalized
standard cochain complex C∗(W,V). When a different transversal is chosen, we get
a different factor set, say f′, such that f −f′ is a coboundary in C2(W,V). So, for
each extension E, there is a unique cohomology class q ∈ H2(W,V) which is called
the extension class of E. A standard result in group extension theory states that up
4
to a suitable equivalence relation on group extensions the converse also holds (see for
example, Theorem 3.12 in Brown [2]):
Proposition 2.1. There is a one-to-one correspondence between equivalence classes of
central extensions of the form E : 0 → V → G → W → 0 and cohomology classes
in H2(W,V) where two extensions E and E′ are considered equivalent if there is a
commutative diagram of the following form:
E : 0 −−−−→ V −−−−→ G −−−−→ W −−−−→ 0
ϕ
(cid:13)  (cid:13)
E′ :0 −−−−→ V(cid:13)(cid:13) −−−−→ Gy′ −−−−→ W(cid:13)(cid:13) −−−−→ 0.
Now, we consider the group theoretical properties of G. Let [, ] denote the com-
mutator [g,h] = g−1h−1gh, and gh denote the conjugation h−1gh. Recall the following
identities which are very easy to verify:
[g,hk] = [g,k][g,h]k
[gh,k] = [g,k]h[h,k] (1)
(gh)2 = g2h2[h,g]h
Applying these to the extension group G, we see that the squaring map induces a map
Q : W → V defined as Q(x)= (xˆ)2, where xˆ denotes an element in G that lifts x∈ W.
Similarly, the commutator induces a symmetric bilinear map B : W ×W → V defined
as B(x,y) = [xˆ,yˆ]. Note also that since V is a central elementary abelian 2-group, the
third row of (1) gives
B(x,y)= Q(x+y)+Q(x)+Q(y)
hence Q is a quadratic map and B is its associated bilinear map.
We now will explain the relation between the quadratic map Q and the extension
class q. Choosing a basis {v ,...,v } for V, we can write Q = (Q ,...,Q ) and
1 n 1 n
B = (B ,...,B ). Also using the isomorphism H2(W,V) ∼= H2(W,Z/2)⊗V, we can
1 n
write q = (q ,...,q ). We have the following:
1 n
Proposition 2.2. Let {w ,...,w } be a basis for W, and let {x ,...,x } denote
1 m 1 m
its dual basis. Under the isomorphism H∗(W,Z/2) ∼= Z/2[x ,...,x ], the cohomology
1 m
class q is equal to the quadratic polynomial associated to Q for each k. In other words,
k k
as a polynomial in x ’s, the cohomology class q is of the following form:
i k
q = Q (w )x2+ B (w ,w )x x .
k k i i k i j i j
Xi Xi<j
Proof. Let E : 0 → V → G → W → 0 be an extension with quadratic map Q and
bilinear map B. We define f :W ×W → V as the bilinear map such that
B(w ,w ) if i < j,
i j
f(w ,w ) = Q(w ) if i = j,
i j i


0 if i > j.


5
It is clear that, beinga bilinear map, f : W×W → V satisfies the factor set conditions.
Also note that for every k, and for every w,w′ ∈ W, we have
f (w,w′) = Q (w )x (w)x (w′)+ B (w ,w )x (w)x (w′),
k k i i i k i j i j
Xi Xi<j
so the associated cohomology class for f is equal to
k
Q (w )x2+ B (w ,w )x x
k i i k i j i j
Xi Xi<j
under the isomorphism H∗(W,Z/2) ∼= Z/2[x ,...,x ].
1 m
To complete the proof, we just need to show that f is a factor set for E. Note that
for every w = a w +···+a w in W, we have
1 1 m m
f(w,w) = a a f(w ,w ) = a Q(w )+ a a B(w ,w )= Q(w).
i j i j i i i j i j
Xi,j Xi Xi<j
Hence, the proof follows from the following lemma:
Lemma 2.3. Let E : 0→ V → G → W → 0 be an extension with quadratic map Q. A
factor set f : W ×W → V is a factor set for E if and only if f(w,w) = Q(w) for all
w ∈ W.
Proof. If f is a factor set for E determined by the splitting s, then
f(w,w) = s(w+w)−1s(w)s(w) = (s(w))2 = Q(w)
holds for all w ∈ W. For the converse, let f be a factor set satisfying f(w,w) = Q(w)
for all w ∈ W. Suppose f′ is a factor set for E. Then, f′′ = f + f′ is a factor
set such that f′′(w,w) = 0 for all w ∈ W. If 0 → V → G′′−π→W → 0 is the
extension with factor set f′′ and transversal t : W → G′′, then for every g ∈ G′′, we
have g2 = [t(π(g))]2 = f′′(π(g),π(g)) = 0. Thus G′′ is of exponent 2, and hence an
elementary abelian 2-group. This implies that the extension 0 → V → G′′−π→W → 0
splits, and therefore f′′ is cohomologous to zero. So, f′ is cohomologous to f, i.e., f′ is
also a factor set for E.
As a consequence of Propositions 2.1 and 2.2, we obtain the following:
Corollary 2.4. Given a quadratic map Q : W → V, there is a unique (up to equiva-
lence) central extension
E(Q) :0 → V → G(Q) → W → 0
with Q(w) = (wˆ)2 for all w ∈ W.
In fact, there is a natural equivalence between the category of quadratic maps Q :
W → V and the category of central extensions of the form E : 0 → V → G → W → 0.
We skip the details of this categorical equivalence, since we do not need this for this
paper.
We end this section with some examples:
6
Example 2.5. Let gl (F ) denote the vector space of n×n matrices over F . Define
n 2 2
Q : gl (F ) → gl (F ) by Q(A)= A+A2. Then one computes
n 2 n 2
Q(A+B)= Q(A)+Q(B)+[A,B]
where[A,B]= AB−BA. ThusQ is a quadraticmap withassociated bilinear map equal
to the Lie bracket [·,·]. We denote this quadratic map by Q .
gl
n
Let K (Z/2m) be the kernel of the map GL (Z/2m) → GL (Z/2) defined by the
n n n
mod 2 reduction of the entries of the given matrix. Observe that
K (Z/8) = {I+2A | A∈ gl (Z/4)}
n n
is a non-abelian group of exponent 4. Consider the mod 4 reduction map
ϕ: K (Z/8) → K (Z/4).
n n
Note that both K (Z/4) and Ker(ϕ) are isomorphic to gl (F ), and that kerϕ is a
n n 2
central subgroup of K (Z/8). So, we get a central extension of the form
n
E : 0 → gl (F ) → K (Z/8) → gl (F ) → 0.
n 2 n n 2
Note that (I + 2A)2 = I + 4(A + A2), and so the associated Q for this extension, is
indeed equal to Q .
gl
n
Example 2.6. IfW isaF -subspaceofgl (F ),suchthatA ∈ W impliesA2 ∈ W,then
2 n 2
Q induces a quadratic map Q : W → W. We call such a subspace W square-closed.
gl
n
Since [A,B] = Q(A + B)+ Q(A)+ Q(B) we have that W is automatically a sub-Lie
algebra of gl (F ). Some examples of subspaces W with this property are:
n 2
(a) W = sl (F ), the matrices of trace zero.
n 2
(b) W = u (F ), the strictly upper triangular matrices.
n 2
In this case we call the corresponding quadratic map Q : W → W a gl-induced
quadratic map.
3 Bockstein closed extensions
We start with some definitions.
Definition 3.1. If q ,...,q are components of q with respect to some basis of V we
1 n
denote by I(Q) the ideal (q ,...,q ) in H∗(W,F ).
1 n 2
It is an easy to see that the ideal I(Q) is indeed independent of the basis chosen for
V, and hence is completely determined by Q.
Definition 3.2. We say the quadratic map Q : W → V is Bockstein closed if I(Q)
is invariant under the Bockstein operator on H∗(W,Z/2). An extension E : 0 →
V → G(Q) → W → 0 is called Bockstein closed if the associated quadratic map Q is
Bockstein closed.
7
SinceI(Q)isanidealgeneratedbyhomogeneouspolynomialsofdegree2,itisclosed
under the higher Steenrod operations, so I(Q) is a Steenrod closed ideal. We will later
use this fact in Section 7 when we are studying effective extensions.
The main examples of Bockstein closed extensions are gl-induced quadratic maps.
We will prove the Bockstein closeness of these extensions at the end of this section. We
first start with an easy observation.
Proposition 3.3. Let Q : W → V be a quadratic map, and let q ∈ H2(W,V) be the
corresponding extension class. Then, Q is Bockstein closed if and only if there is a one
dimensional class L ∈ H1(W,End(V)) such that β(q) = Lq. Here, the multiplication
Lq is given by the composite
H1(W,End(V))⊗H2(W,V)−∪→H3(W,End(V)⊗V)→ H3(W,V)
where the second map is induced by the evaluation map ev : End(V)⊗V → V, given
by f ⊗u→ f(u).
Proof. ChooseabasisB forW andB forV,andlet{x ,...,x }bethedualbasisto
W V 1 m
B . We can writeq as a column vector with entries in H2(W,F ). Then,q is Bockstein
W 2
closed if and only if there is an n×n matrix L with entries in linear polynomials in x ’s
i
such that β(q) = Lq where n = dim(V). Note that L can be considered as an element
in H1(W,M (F )). Using the basis B , we can also identify M (F ) with End(V).
n×n 2 V n×n 2
So, L can be considered as a class in H1(W,End(V)), and it is easy to see that under
these identifications, matrix multiplication Lq corresponds to the map given above.
Definition 3.4. Let Q : W → V be a quadratic map, and let q ∈ H2(W,V) be the
corresponding extension class. If L ∈ H1(W,End(V)) is a class satisfying β(q) = Lq,
then we say L is a binding operator for Q. If a specific basis for V is chosen then L can
represented as a matrix with linear polynomial entries. In this case we call L a binding
matrix.
It is clear that one can have many different matrices for the same operator. Any
two such matrices will be conjugate to each other by a scalar matrix. On the other
hand, one can also have two different binding operators L and L for a quadratic map
1 2
Q. The following example shows a case where this happens.
Example 3.5. Let q = (xy,yz). Then, β(q) = (xy(x+y),yz(y+z)). We can write
β(q ) x+y 0 q
1 = 1
(cid:20)β(q2)(cid:21) (cid:20) 0 y+z(cid:21)(cid:20)q2(cid:21)
or
β(q ) x+y+z x q
1 = 1 .
(cid:20)β(q2)(cid:21) (cid:20) z x+y+z(cid:21)(cid:20)q2(cid:21)
Note that the second binding matrix is not conjugate to a diagonal matrix.
This can happen because the ideal (q ,q ) is not a free k[x,y,z]-module over the
1 2
generators {q ,q }. We will see later that under stronger conditions there is a unique
1 2
binding operator L for q.
The following is a reformulation of Proposition 3.3 which is quite useful in many
instances.
8
Theorem 3.6. Let Q : W → V be a quadratic map, and let B be the bilinear map
associated to Q. Then, Q is Bockstein closed if and only if there exists a bilinear map
P :V ×W → V which satisfies the identity
P(Q(x),y) = B(x,y)+P(B(x,y),x)
for all x,y ∈ W.
Before the proof we first do some calculations. Let {v ,...,v } be a basis for V,
1 m
and let {w ,...,w } be a basis for W with dual basis {x ,...,x }. As usual we write
1 n 1 n
q = (q ,...,q ) for the extension class q ∈ H2(W,V) associated to Q. Note that by
1 n
Proposition 2.2, for each k = 1,...,m, we have
q = Q (i)x2 + B (i,j)x x
k k i k i j
Xi Xi<j
where Q (i) = Q (w ) and B (i,j) = B (w ,w ). Applying the Bockstein we get
k k i k k i j
β(q ) = B (i,j) x2x +x x2 = B (i,j)x2x
k k i j i j k i j (2)
Xi<j (cid:0) (cid:1) Xi,j
Given an element L ∈ H1(W,End(V)), we can consider it as a homomorphism
L : W → End(V) via the isomorphism
H1(W,End(V) ∼= Hom(W,End(V)).
This allows us to describe a correspondence between classes L ∈ H1(W,End(V)) and
bilinearmapsP : V×W → V. ThecorrespondenceisgivenbyP(v,w) = L(w)(v)forall
v ∈ V andw ∈ W. Therefore,if Ldenotes thematrix for anelement in H1(W,End(V))
with respect to the above choice of basis for V and W, then we can write
L = P (s,p)x
ks k p
Xp
where P (s,p) is short for P (v ,w ) and P is the k-th coordinate of P. We have the
k k s p k
following:
Lemma 3.7. The equation β(q) = Lq holds if and only if the following two equations
hold
P(Q(w ),w ) = B(w ,w )+P(B(w ,w ),w ) (3)
i j i j i j i
P(B(w ,w ),w )+P(B(w ,w ),w )+P(B(w ,w ),w ) = 0 (4)
i j k j k i k i j
for all i,j,k ∈ {1,...,n}.
Proof. The proof is an easy consequence of carefully writing down the equation β(q) =
Lq in terms of the variables and comparing the coefficients.
First note that by Equation 2 above, we have
β(q ) = B (i,j)x2x
k k i j
Xi,j
9
for all k. On the other hand, the k-th entry of Lq is equal to L q . Writing this
s ks s
sum in detail, we get P
L q = P (s,p)x Q (i)x2+ B (i,j)x x
ks s k p s i s i j
Xs Xs h(cid:16)Xp (cid:17)(cid:16)Xi Xi<j (cid:17)i
= Q (i)P (s,p)x2x + B (i,j)P (s,p)x x x
s k i p s k p i j
Xs Xp,i Xs pX,i<j
= Q (i)P (s,p) x2x + B (i,j)P (s,p) x x x
s k i p s k p i j
Xp,i(cid:0)Xs (cid:1) pX,i<j(cid:0)Xs (cid:1)
= Q (i)P (s,j) x2x + B (i,j)P (s,p) x x x
s k i j s k p i j
Xi,j (cid:0)Xs (cid:1) pX,i<j(cid:0)Xs (cid:1)
The second term on the righthand side (SLHS) can be manipulated more:
SLHS = B (i,j)P (s,i) x2x + B (i,j)P (s,j) x x2
s k i j s k i j
pX=i<j(cid:0)Xs (cid:1) i<Xj=p(cid:0)Xs (cid:1)
+ B (i,j)P (s,p) x x x
s k p i j
Xi<j (cid:0)Xs (cid:1)
p6=i,j
= B (i,j)P (s,i) x2x + B (i,j)P (s,p) x x x
s k i j s k p i j
Xi,j (cid:0)Xs (cid:1) Xi<j (cid:0)Xs (cid:1)
p6=i,j
where the last line follows from the identity B (i,j) = B (j,i). Putting these equations
s s
together, we get
β(q )+ L q = B (i,j)+ Q (i)P (s,j)+ B (i,j)P (s,i) x2x
k ks s k s k s k i j
Xs Xi,j (cid:16) Xs Xs (cid:17)
+ B (i,j)P (s,p) x x x
s k p i j
Xi<j (cid:0)Xs (cid:1)
p6=i,j
Fixing an order for {p,i,j}, the second summand in the above equation becomes
B (i,j)P (s,p)+B (p,j)P (s,i)+B (p,i)P (s,j) x x x
s k s k s k p i j
pX<i<jhXs (cid:16) (cid:17)i
So, we obtain that β(L)+L2 = 0 if and only if
B (i,j)+ Q (i)P (s,j)+ B (i,j)P (s,i) = 0
k s k s k
Xs Xs
and
B (i,j)P (s,p)+B (p,j)P (s,i)+B (p,i)P (s,j) = 0
s k s k s k
Xs (cid:16) (cid:17)
holds forall i,j,k. Itis easy to see thattheseequations arethesame as theEquations 3
and 4.
We will also need the following lemma:
10