Right Coideal Subalgebras of the Quantum Borel Algebra of type G2

0.24 MB·English

by  Barbara Pogorelsky

Right Coideal Subalgebras of the Quantum Borel
Algebra of type G ∗
0 2
1
0
2 B´arbara Pogorelsky

n [email protected]
a
InstitutodeMatema´tica, UniversidadeFederaldoRioGrandedoSul
J
Av. BentoGon¸calves 9500, PortoAlegre,RS,91509-900, Brazil
7

January 7, 2010
]
A
Q
.
h Abstract
t
a
In this paper we describe the right coideal subalgebras containing all
m
group-like elements of the multiparameter quantum group U+(g), where
q
[ g is a simple Lie algebra of type G2, while the main parameter of quan-
tization q is not a root of 1. If the multiplicative order t of q is finite,
1
v t > 4, t 6=6, then the same classification remains valid for homogeneous
0 rightcoidealsubalgebrasofthepositivepartu+q(g)ofthemultiparameter
2 version of thesmall Lusztig quantumgroup.
1
1 Keywords: Quantumgroups,Hopfalgebras,Coidealsubalgebras,PBW-generators
. MathematicsSubjectClassification: 16W30,17B37
1
0
0 1 Introduction
1
:
v ComodulealgebrasoveraHopfalgebranaturallyariseintheGaloistheoryof
i
X HopfalgebraactionsastheGaloisobjects;seeA.MasuokaandT.Yanai[15],A.
Milinski[16],S.WestreichandT.Yanai[19]andT.Yanai[20,21]. Inparticular
r
a the Galois correspondence theorem for the actions on a free algebra sets up a
one to one correspondence between all right coideal subalgebras and all inter-
mediate free subalgebras; see V.O. Ferreira, L.S.I. Murakami, and A. Paques
[4]. At the same time, the notion of one-sided coideal subalgebras appears to
be of fundamental importance in the theory of quantum groups: a survey by
G. Letzter [14] provides anoverviewof the use of one-sidedcoidealsubalgebras
inconstructingquantumsymmetricpairs,informingquantumHarish-Chandra
modules and in producing quantum symmetric spaces.
∗ThispaperwaswrittenduringaresearchperiodattheUNAMFES-C,Mexico,withthe
supportofCNPq-Brazil. Itwillbepartoftheauthor’sPhDthesis.
1
RecentlyV.K.KharchenkoandA.V.LaraSagah´on[12],usingaPBW-basis
construction method [11], offered a complete classification of right coideal sub-
algebrasthatcontainthecoradicalk[G]forthequantumgroupU (sl ). Asa
q n+1
consequencetheydeterminedthatthequantumBorelalgebraU+(g),g=sl ,
q n+1
contains (n+1)! different right coideal subalgebras that include the coradical.
If g = so is a simple Lie algebra of type B then U+(g) has (2n)!! right
2n+1 n q
coideal subalgebras that include the coradical [13]. In both cases the number
coincides with the order of the Weyl group defined by the Lie algebra g. This
providesenoughreasontoconjecturethatforarbitrarysimplefinitedimensional
Lie algebras g the number of right coideal subalgebras in U+(g) that include
q
the coradicalcoincides with the order of the Weyl group related to g, see [13].
In this paper by means of the same PBW-basis construction method we
provethis conjecture for the Lie algebrag of type G . More precisely,we prove
2
the following theorem.
Theorem 1.1. If q is not a root of 1, the lattice of right coideal subalgebras
containing k[G] of U+(g) is given in the Figure 1. If q has multiplicative or-
q
der t > 4, t 6= 6, the same figure is the lattice of homogeneous right coideal
subalgebras containing k[G] of u+(g).
q
U+(g)
q
h[[x ,x ],[x ,[x ,x ]]]i h[x ,x ]i
1 2 2 2 1 1 2
h[x ,[x ,[x ,x ]]]i h[[x ,x ],[[x ,x ],x ]]i
2 2 2 1 1 2 1 2 2
h[x ,[x ,x ]]i h[[x ,x ],x ]i
2 2 1 1 2 2
h[x ,x ]i h[[[x ,x ],x ],x ]i
2 1 1 2 2 2
hx i hx i
1 2
k[G]
Figure 1: Lattice of Right Coideal Subalgebras
Herethe distinguishedelementisageneratorofthe rightcoidealsubalgebra
(in particular eachright coidealsubalgebrais generatedoverthe coradicalby a
single element).
2
In the second section, following to [12], we introduce main concepts and
generalresultsthatareofuseforfurtherconsiderations. Inthethirdsectionwe
considerthe algebraU+(g)(respectively,u+(g))asacharacterHopfalgebra[9]
q q
in order to find a PBW-basis in the explicit form (Theorem 3.1). The results
of this section are very similar to the results of the section 4.2 of the paper [3]
by I. Angiono dedicated to classification of finite dimensional Nichols algebras
(quantum symmetric algebras) over algebraically closed fields of characteristic
zero. In the fourth section, following [11], we transform the found PBW-basis
up to a PBW-basis of a given right coideal subalgebra. In this way we may
find all possible PBW-bases for rightcoidealsubalgebrascontaining k[G]. This
provides the required classification (Theorem 1.1).
2 Preliminaries
Definition 2.1. Let S be an algebra over a field k and A its subalgebra with
a fixed basis {a |j ∈J}. A linearly orderedsubset W ⊆S is said to be a set of
j
PBW-generators of S over A if there exists a function h:W →Z+∪∞, called
the height function, such that the set of all products
a wn1wn2...wnk, (2.1)
j 1 2 k
where j ∈ J, w < w < ... < w ∈ W, n < h(w ), 1 ≤ i ≤ k is a basis of S.
1 2 k i i
The value h(w) is referred to as the height of w in W. If A = k is the ground
field, then we shall call W simply as a set of PBW-generatorsof S.
Definition 2.2. Let W be a set of PBW-generators of S over a subalgebra
A. Suppose that the set of all words in W as a free monoid has its own order
≺ (that is, a ≺ b implies cad ≺ cbd for all words a,b,c,d ∈ W). A leading
word of s ∈ S is the maximal word m = wn1wn2...wnk that appears in the
1 2 k
decomposition of s in the basis (2.1). A leading term of s is the sum am of all
terms α a m, α ∈ k, that appear in the decomposition of s in the basis (2.1),
i i i
where m is the leading word of s.
Definition 2.3. AHopfalgebraH issaidacharacter Hopf algebra ifthegroup
G of all group-like elements is commutative and H is generated over k[G] by
skew primitive semi-invariants a ,i∈I:
i
∆(a )=a ⊗1+g ⊗a , g−1a g =χi(g)a , g,g ∈G,
i i i i i i i
where χi, i∈I, are characters of the group G.
Let us associate a quantum variable x to a . For each word u in X =
i i
{x |i ∈ I} we denote by g an element of G that appears from u by replacing
i u
each x with g . In the same way we denote by χu a character that appears
i i
from u by replacing each x with χi. We define a bilinear skew commutator on
i
homogeneous linear combinations of words by the formula
[u,v]=uv−χu(g )vu, (2.2)
v
3
where we use the notation χu(g ) = p = p(u,v). These brackets are related
v uv
to the product by the following identities
[u·v,w]=p [u,w]·v+u·[v,w], (2.3)
vw
[u,v·w]=[u,v]·w+p v·[u,w]. (2.4)
uv
ThegroupGactsonthe freealgebrakhXibyg−1ug =χ(g)u,whereuisan
arbitrary monomial in X. The skew group algebra GhXi has the natural Hopf
algebra structure
∆(x )=x ⊗1+g ⊗x , i∈I, ∆(g)=g⊗g.
i i i i
We fix a Hopf algebra homomorphism
ξ :GhXi→H, ξ(x )=a , ξ(g)=g, i∈I, g ∈G.
i i
Definition 2.4. Aconstitution ofaworduinG∪X isafamilyofnon-negative
integers{m ,x∈X}suchthatuhas m occurrencesofx. Certainlyalmostall
x x
m in the constitution are zero.
x
Letusfixanarbitrarycompleteorder<onthesetX,andletΓ+ bethefree
additive(commutative)monoidgeneratedbyX. ThemonoidΓ+isacompletely
ordered monoid with respect to the following order:
′ ′ ′
m x +m x +...+m x >m x +m x +...+m x (2.5)
1 i1 2 i2 k ik 1 i1 2 i2 k ik
if the first from the left nonzero number in (m −m′,m −m′,...,m −m′)
1 1 2 2 k k
is positive, where x > x > ... > x in X. We associate a formal degree
i1 i2 ik
D(u) = m x ∈ Γ+ to a word u in G∪X, where {m |x ∈ X} is the
Px∈X x x
constitution of u. Respectively, if f = α u ∈GhXi, 06=α ∈k then
P i i i
D(f)=max {D(u )}. (2.6)
i i
On the set of all words in X we fix the lexicographical order with the priority
from the left to the right, where a proper beginning of a word is considered to
be greater than the word itself.
Definition 2.5. A non-empty word u is called a standard word (or Lyndon
word, or Lyndon-Shirshov word) if vw > wv for each decomposition u = vw
with non-empty v,w.
Definition 2.6. Anon-associative word is awordwherebrackets[,]aresome-
how arrangedto show how multiplication applies.
If [u] denotes a non-associative word, then by u we denote an associative
wordobtainedfrom[u]by removingthe brackets. Ofcourse,[u]is notuniquely
defined by u in general.
Definition 2.7. The set of standard non-associative words is the biggest set
SL that contains all variables x and satisfies the following properties:
i
4
1. If [u]=[[v],[w]]∈SL then [v],[w]∈SL, and v >w are standard.
2. If [u]=[[[v ],[v ]],[w]]∈SL then v ≤w.
1 2 2
By the Shirshov’s Theorem, every standard word has only one alignment of
brackets such that the defined non-associative word is standard. In order to
find this alignment we use the following procedure: the factors v,w of the non-
associative decomposition [u] = [[v],[w]] are standard words such that u = vw
and v has the minimal length (see [18]).
Definition 2.8. A super-letter is a polynomial that equals a non-associative
standardwordwherethebracketsmean(2.2). Asuper-word isawordinsuper-
letters.
Using Shirshov’s Theorem, every standard word u defines only one super-
letter that will be denoted by [u]. The order on the super-letters is defined in
the natural way: [u]>[v]⇔u>v.
Since quantum Borel algebras U+(g) and u+(g) are homogeneous in each
q q
variable, in what follows we suppose that H is a Γ+-graded character Hopf
algebra, that is, H is homogeneous in each of the generators a .
i
Definition 2.9. A super-letter [u] is called hard in H if its value in H is not
a linear combination of super-words of the same degree (2.6) in super-letters
smaller than [u].
Proposition 2.10. ([7,Corollary2])A super-letter [u] is hard in H if and only
if the value in H of the standard word u is not a linear combination of values
of smaller words of the same degree (2.6).
Proposition 2.11. ([9, Lemma 4.8]) Let B be a set of super-letters contain-
ing x ,...,x . If each pair [u],[v] ∈ B, u > v satisfies one of the following
1 n
conditions
1) [[u],[v]] is not a standard non-associative word;
2) the super letter [[u],[v]] is not hard in H;
3) [[u],[v]]∈B;
then the set B includes all hard in H super-letters.
Definition 2.12. We say that the height of a hardin H super-letter [u]equals
h=h([u]) if h is the smallest number such that
1. p is a primitive t-th root of 1 and either h = t or h = tlr, where
uu
l =char(k),
2. the value of [u]h in H is a linear combination of super-words of the same
degree (2.6) in super-letters smaller than [u].
If there exists no such number then the height equals infinity.
5
Theorem 2.13. ([7, Theorem 2]) The values of all hard in H super-letters
with the above defined height function form a set of PBW-generators for H over
k[G].
According to [11, Theorem 1.1], every right coideal subalgebra U that con-
tains allgroup-likeelements has a PBW-basisoverk[G] whichcanbe extended
up to a PBW-basis of H. The PBW-generators T for U can be obtained from
the PBW-basis of H given in Theorem 2.13 in the following way.
Suppose that for a given hardsuper-letter [u] there exists an element c∈U
c=[u]s+XαiWi (2.7)
whereW arethe basissuper-wordsstartingwithsuper-letterssmallerthan[u],
i
D(W ) = sD(u). We fix one of the elements with minimal s and denote it by
i
c . Thus, for every super-letter [u] hard in H we have at most one element c .
u u
We define the height function by the following lemma.
Lemma 2.14. ([11, Lemma 4.3]) In the representation (2.7) of the chosen
element c , either s=1 or p(u,u) is a primitive t-th root of 1 and s=t, or (in
u
the case of positive characteristic) s=t(chark)r.
Iftheheightof[u]inH isinfinite,thentheheightofc inUisdefinedtobe
u
infinite as well. If the height of [u] in H equals t and p(u,u) is a primitive t-th
root of 1, then, due to the above lemma, s=1 (note that in the representation
(2.7) the number s is less than the height of [u]). In this case, the height of c
u
inUissupposedtobetaswell. Ifthecharacteristicl ispositiveandtheheight
of [u] in H equals tlr, then we define the height of c in U to be equal to tlr/s
u
(thus,incharacteristiczerotheheightofc inUalwaysequalstheheightof[u]
u
in H).
Proposition 2.15. ([11, Proposition 4.4]) The set of all chosen c with the
u
above defined height function forms a set of PBW-generators for U over k[G].
Definition 2.16. (see, for example, [12, section 3]) Let C =k a k be a gen-
ij
eralized Cartan matrix symmetrizable by D = diag(d ,...,d ), d a = d a .
1 n i ij j ji
Denote by g a Kac-Moody algebra defined by C (see [5]). Suppose that the
quantification parameters p =p(x ,x )=χi(g ) are related by
ij i j j
p =qdi, p p =qdiaij, 1≤i,j ≤n. (2.8)
ii ij ji
ThemultiparameterquantizationU+(g)oftheBorelsubalgebrag+isacharacter
q
Hopf algebra generated by x ,...,x ,g ,...,g and defined by Serre relations
1 n 1 n
with the skew brackets (2.2) in place of the Lie operation:
[[...[[x ,x ],x ],...],x ]=0, 1≤i6=j ≤n, (2.9)
i j j j
where x appears 1−a times.
j ji
By [6, Theorem6.1] the left hand sides of these relationsare skew-primitive
elements in GhXi. Therefore, the ideal generated by these elements is a Hopf
ideal, while U+(g) indeed has a natural character Hopf algebra structure.
q
6
Definition 2.17. (see, for example, [12, section 3]) If the multiplicative order
t of q is finite, then we define u+(g) as GhXi/Λ, where Λ is the biggest Hopf
q
ideal in GhXi(2), which is the set (an ideal) of noncommutative polynomials
without free and linear terms. From [10, Lemma 2.2]this is a Γ+-homogeneous
ideal. Certainly Λ contains all skew-primitive elements of GhXi(2) (each one of
them generates a Hopf ideal). Hence, by [6, Theorem 6.1], relations (2.9) are
still valid in u+(g).
q
We notice that the subalgebra A generated by x ,...,x over k in U+(g)
1 n q
is a Nichols algebra of Cartan type if q is not a root of 1, as described in
[1]. Analogously, if qt = 1 for an integer t, the same thing is valid for A ⊆
u+(g). This is particularly useful since in [3] there are many results for the
q
Nichols algebra A, although in this paper k is an algebraically closed field of
characteristic zero. However, if q is a root of 1, then the subalgebra generated
by x ,...,x in U (g) is not a Nichols algebra.
1 n q
Definition 2.18. The subalgebra A generated by x ,...,x over k in U+(g)
1 n q
(respectively, u+(g)) has a differential calculus defined by
q
∂ (x )=δj, ∂ (uv)=∂ (u)v+p(u,x )u∂ (v),
i j i i i i i
for x ∈X.
i
Lemma 2.19. ([13, Lemma 2.10]) Let u∈ khXi be a homogeneous in each x
i
element. If p is a t-th primitive root of 1, then
uu
∂ (ut)=p(u,x )t−1[u,[u,...[u,∂ (u)]...]].
i i i
| t{−z1 }
Lemma 2.20. (Milinski-Schneidercriterion,see[17])If a polynomial f ∈ khXi
with no free term is such that ∂ (f)=0 in u+(g) for every x ∈X, then f =0
i q i
in u+(g).
q
3 Explicit PBW-Generators for Quantizations
In this section we are going to explicit a set of PBW-generators for U+(g)
q
(respectively,u+(g),ifqt =1fort>4,t6=6),wheregisthesimpleLiealgebra
q
of type G .
2
Let us first remember that the algebra U+(g) is defined by two generators
q
x ,x and two relations
1 2
[[[[x ,x ],x ],x ],x ]=0, [x ,[x ,x ]]=0, (3.1)
1 2 2 2 2 1 1 2
where the brackets mean the skew commutator (2.2). Relations (2.8) take up
theformp =p3 ,p p =p−1,andp =q. Inwhatfollowsweshallsuppose
11 22 12 21 11 22
that q6 6=1 and q4 6=1.
7
Using (2.2), the relations in (3.1) can also be written as
x x4+a x x x3+a x2x x2+a x3x x +a x4x =0, (3.2)
1 2 1 2 1 2 2 2 1 2 3 2 1 2 4 2 1
x2x +b x x x +b x x2 =0, (3.3)
1 2 1 1 2 1 2 2 1
where
a =−p p[4], a =p2 p p[3](p2 +1), a =−p3 p3 p[4], a =p4 p6 ,
1 12 22 2 12 22 22 22 3 12 22 22 4 12 22
b =−p (1+p ), b =p2 p ,
1 12 11 2 12 11
andasusualwedenote p[n] =1+p+...+pn−1 (anotherusualnotationis [n] ).
p
Let us multiply (3.2) from the left by x , while (3.3) from the right by x3.
1 2
The difference of the obtained relations provides a new one
(a −b )x x x x3 =−a x x2x x2−a x x3x x −a x x4x +b x x2x3. (3.4)
1 1 1 2 1 2 2 1 2 1 2 3 1 2 1 2 4 1 2 1 2 2 1 2
We may multiply (3.2) from the left by (a −b )x x , while (3.4) from the
1 1 1 2
right by x . Again the difference gives a new relation
2
{a (a −b )−a }x x2x x3 ={a −a (a −b )}x x3x x2
1 1 1 2 1 2 1 2 3 2 1 1 1 2 1 2
+{a −a (a −b )}x x4x x −a (a −b )x x5x −b x x2x4. (3.5)
4 3 1 1 1 2 1 2 4 1 1 1 2 1 2 2 1 2
In the same way, if we multiply (3.3) from the right by (a −b )x x3, while
1 1 1 2
(3.4) from the left by x , the difference of the obtained relations after replace-
1
ment of all subwords x2x with −b x x x −b x x2 defines a new relation
1 2 1 1 2 1 2 2 1
(a b −{b (a −b )+b }b )x x x x x x2 ={b (a −b )+b }b x x2x2x2
2 1 1 1 1 2 1 1 2 1 2 1 2 1 1 1 2 2 1 2 1 2
−a b x x x x2x x −a b x x x x3x +W. (3.6)
3 1 1 2 1 2 1 2 4 1 1 2 1 2 1
where W is a linear combination of words with the first letter x .
2
Now we are ready to prove the following statement.
Proposition 3.1. All hard in U+(g) (respectively, in u+(g)) super-letters are
q q
contained in the following list:
[A]=x ,
1
[B]=[x ,x ],
1 2
[C]=[[x ,x ],[[x ,x ],x ]], (3.7)
1 2 1 2 2
[D]=[[x ,x ],x ],
1 2 2
[E]=[[[x ,x ],x ],x ],
1 2 2 2
[F]=x .
2
Proof. To prove the proposition we’ll show that this set satisfies the condi-
tions of Proposition 2.11, that is, for every pair [X],[Y] in this set such that
X >Y,the non-associativeword[[X],[Y]]eitherbelongstothisset,oritisnot
astandardnon-associativeword,orit definesa non-hardinU+(g) super-letter.
q
8
Ifoneofthese3possibilitiesissatisfiedforeverypossiblepair,thenProposition
2.11 proves that all the hard in U+(g) super-letters are contained in the list
q
(3.7).
We have 15 possibilities. In four cases [[A],[F]] = [B], [[B],[D]] = [C],
[[B],[F]] = [D] and [[D],[F]] = [E]. In five more cases ([[A],[B]], [[A],[C]],
[[A],[D]],[[A],[E]]and[[E],[F]])thenon-associativewordisnothardbyPropo-
sition2.10,sincetheassociativewordobtainedbyomittingthebracketscontains
either subword x x4 or subword x2x , and therefore by the relations (3.2) and
1 2 1 2
(3.3)itisalinearcombinationofsmallerwordsinU+(g). Intwocases,[[C],[E]]
q
and [[C],[F]], the word is not standard as a non-associative word. It remains
to consider the following four cases:
• [[D],[E]]: the relation (3.5) shows that the word DE is a linear combina-
tionofsmallerwordsinU+(g). Thus,byProposition2.10,thesuper-letter
q
is not hard.
• [[B],[C]]: since BC = (x x )3x , the relation (3.6) shows that the word
1 2 2
BC is a linear combination of smaller words. Again, by Proposition2.10,
the super-letter is not hard.
• [[B],[E]]: in this caseBE =x x x x3, andwemayuse the relation(3.4).
1 2 1 2
• [[C],[D]]: let us multiply the relation (3.6) from the right by (a −b )x ,
1 1 2
while the relation(3.4) fromthe left by (a b −{b (a −b )+b }b )x x .
2 1 1 1 1 2 1 1 2
The leading term of the difference equals
{a b (a −b )−a (a b −{b (a −b )+b }b )}x x x x2x x2
3 1 1 1 2 2 1 1 1 1 2 1 1 2 1 2 1 2
=−p5 p5 (1+p3 )(1+p2 )CD.
12 22 22 22
Therefore CD is also a linear combination of smaller words.
Thus, by Proposition 2.11, the set {[A],[B],[C],[D],[E],[F]} contains all
hard in U+(g) super-letters.
q
Sinceu+(g)isahomomorphicimageofU+(g),allnon-hardinU+(g)super-
q q q
letters are non-hard in u+(g). Hence the list (3.7) contains all hard in u+(g)
q q
super-letters as well. (cid:3)
Here we have to make an important remark. A PBW-basis over k for the
Nichols algebra of Cartan type G is computed in [3], and as a consequence we
2
have a PBW-basis over k[G] for U+(g) provided that q is not a root of 1, and
q
foru (g)otherwise. However,thepreviouspropositionisnecessarysincetofind
q
the PBW-generators for the possible right coideal subalgebras U we need not
just a PBW-basis, which is not unique, but the exact PBW-basis constituted
by the hard super-letters, as described in Theorem 2.13 and Proposition 2.15.
Note that, according to Definition 2.18, the subalgebra A = khx ,x i of
1 2
U+(g) (respectively, u+(g)) has a differential calculus
q q
∂ (x )=δj, ∂ (uv)=∂ (u)·v+p(u,x )u·∂ (v) (3.8)
i j i i i i i
9
for i=1,2.
Now we notice that, if ∂ (u)=0, then:
i
∂ ([u,x ])=∂ (u)·x +p(u,x )u·∂ (x )−p(u,x )(∂ (x )·u+p(x ,x )x ·∂ (u))=0.
i i i 2 i i i i i i i i i i
(3.9)
From [12, Lemma 2.10] we know that the homogeneous right coideal sub-
algebras containing k[G] have the form U = U #k[G], where U = A∩U
A A
is a differential subalgebra of A = khx ,x i and G is the set of all grouplike
1 2
elements. So, from knowing the differential subalgebras, we can describe the
homogeneous right coideal subalgebras.
Using the formulas (3.8) and (3.9) we have the following result.
Proposition 3.2. The derivatives of the elements from the list (3.7) are given
in the Figure 2.
∂ ∂
1 2
[A] 1 0
[B] (1−q−3)x 0
2
[C] q2(1−q−3)2x [D]+p (1−q−3)(q3−q2−q)[E] 0
2 21
[D] (1−q−3)(1−q−2)x2 0
2
[E] (1−q−3)(1−q−2)(1−q−1)x3 0
2
[F] 0 1
Figure 2: Table of Derivatives
Proof. Since [A] = x and [F] = x , from the definition, ∂ ([A]) = 1,
1 2 1
∂ ([A])=0, ∂ ([F])=0, ∂ ([F])=1.
2 1 2
For [B] we have [B]=[x ,x ]=x x −p x x . Using (3.8),
1 2 1 2 12 2 1
∂ ([B])=∂ (x x )−p ∂ (x x )=
1 1 1 2 12 1 2 1
=∂ (x )x +p x ∂ (x )−p (∂ (x )x +p x ∂ (x ))=
1 1 2 11 1 1 2 12 1 2 1 21 2 1 1
=x −p p x =(1−q−3)x .
2 12 21 2 2
For [D]=[[x ,x ],x ]=[[B],x ]=[B]x −p p x [B], we have
1 2 2 2 2 12 22 2
∂ ([D])=∂ ([B]x )−p p ∂ (x [B])=
1 1 2 12 22 1 2
=∂ ([B])x +p p [B]∂ (x )−p p (∂ (x )[B]+p x ∂ ([B]))=
1 2 11 21 1 2 12 22 1 2 21 2 1
=(1−p p )x2+0−0−p p p (1−p p )x2 =
12 21 2 12 22 21 12 21 2
=(1−p p p )(1−p p )x2 =(1−q−3)(1−q−2)x2.
12 22 21 12 21 2 2
Again, for [E]=[[[x ,x ],x ],x ]=[[D],x ]=[D]x −p p2 x [D], we have
1 2 2 2 2 2 12 22 2
∂ ([E])=∂ ([D]x )−p p2 ∂ (x [D])=
1 1 2 12 22 1 2
=∂ ([D])x +p p2 [D]∂ (x )−p p2 (∂ (x )[D]+p x ∂ ([D]))=
1 2 11 21 1 2 12 22 1 2 21 2 1
=(1−p p2 p )(1−p p p )(1−p p )x3 =
12 22 21 12 22 21 12 21 2
=(1−q−3)(1−q−2)(1−q−1)x3.
2
10