A matrix realization of the quantum group g_{p, q}

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by  Yusuke Arike

A MATRIX REALIZATION OF THE QUANTUM GROUP g
p,q
YUSUKE ARIKE
Abstract. In this paper we will find a matrix realizations of the quantum group g
p,q
defined in [4]. For this purpose, we construct all primitive idempotents and a basis
0
of g . We determine the action of elements of the basis on the indecomposable
1 p1,p2
0 projective modules, which give rise to a matrix realization of gp1,p2. By using this
2 result, we obtain a basis of the space of symmetric linear functions on g and express
p,q

n the symmetric linear functions obtained by the left integral, the balancing element and
a
the center of g in term of this basis.
J p,q

3
2

]
A 1. Introduction
Q
In [2] and [3], it is pointed out that the triplet vertex operator algebra (p) is closely
.
h W
t related with the restricted quantum group Uq(sl2) at the primitive 2p-th root of the unity.
a
m It is also foundin [2] and[3] findthat the space of functions obtained by applying modular
[ transformations to characters of simple modules of (p) is isomorphic to the center of
W
U (sl ). It is conjectured in [3] that the category of modules of (p) and the category of
2 q 2
W
v finite-dimensional modules of U (sl ) are equivalent as tensor categories. It is proved the
q 2
1
3 conjecture for p = 2. The equivalence of categories as abelian categories is proved in [10].
3 We also mention that the tensor category consisting of finite-dimensional U (sl )-modules
0 q 2
. is not braided if p = 2 ([7]).
4 6
0 Feigin, Gainutdinov, Semikhatov and Tipunin also construct the vertex operator alge-
9 bra (p ,p ) for coprime positive integers p and p . They find 2p p + 1(p 1)(p 1)
0 W 1 2 1 2 1 2 2 1− 2−
: non-isomorphic simple modules and show that the space of functions obtained by ap-
v
i plying modular transformations on characters of simple modules is 1(3p 1)(3p 1)-
X 2 1 − 2 −
dimensional. They study in [4] the unimodular finite-dimensional Hopf algebra g
r p1,p2
a whichisaquotientalgebraofthetensorproductoftworestrictedquantumgroupsU (sl )
q1 2
at q = exp(√ 1p π/p ) and U (sl ) at q = exp(√ 1p π/p ). In [5], they show that
1 − 2 1 q2 2 1 − 1 2
g has 2p p non-isomorphic simple modules, which shows that the category of mod-
p1,p2 1 2
ules of (p ,p ) is not equivalent to the category of finite-dimensional modules of g .
W 1 2 p1,p2
They prove that the space of functions obtained by applying modular transformations to
characters of simple modules of (p ,p ) and the center of g , on which the group
W 1 2 p1,p2
SL (Z) naturally acts, are equivalent as representations of SL (Z).
2 2
In this paper, we determine a matrix realization of g . More precisely, it is shown
p1,p2
in [5] that the quantum group g has a decomposition into two-sided ideals
p1,p2
p1 1 p2 1
− −
(1.0.1) g = Q(p ,p ) Q(0,p ) Q(r ,p ) Q(p ,r ) Q(r ,r ),
p1,p2 1 2 ⊕ 2 ⊕ 1 2 ⊕ 1 2 ⊕ 1 2
rM1=1 rM2=1 (r1M,r2)∈I
2000 Mathematics Subject Classification. Primary 16W35, Secondary 17B37, 81R05.
1
2 Y. ARIKE
where I = (r ,r ) 1 r p 1, p r +p r p p , we will get a matrix realization of
1 2 i i 1 2 2 1 1 2
{ | ≤ ≤ − ≤ }
atwo-sided ideal which appears in(1.0.1). Inorder toconstruct a basis, we first determine
primitive idempotents and derive direct sum decompositions of two-sided ideals into their
indecomposable left ideals as it is done in [1] and [12]. Then, by calculating actions
of two-sided ideals on projective modules, we can have matrix realizations of two-sided
ideals.
Theorem 6.1. The algebras Q(p ,p ) and Q(0,p ) are isomorphic to the matrix algebras
1 2 2
M (C).
p1p2
Theorem 6.2. For 1 r p 1, the algebra Q(r ,p ) is isomorphic to the subalgebra
1 1 1 2
≤ ≤ −
of M (C) M (C) with the shape:
2p1p2 ⊕ 2p1p2
Br+1,,↑p2 0 0 0 Bp−1,↑r1,p2 0 0 0
BBrr++11,,,,→←pp22 Bp−1,−↑0r1,p2 Bp−1,↑0r1,p2 00 ,BBpp−−11,,−−←→rr11,,pp22 Br+01,,↑p2 Br+01,,↑p2 00  .
(cid:16)Br+1,,↓p2 Bp−1,→r1,p2 Bp−1,−←r1,p2 Br+1,,↑p2 Bp−1,−↓r1,p2 Br+1,,→p2 Br−1,,←p2 Bp−1,↑r1,p2(cid:17)
 − −   − − 
Theorem 6.3. For 1 r p 1, the algebra Q(r ,p ) is isomorphic to the subalgebra
1 1 1 2
≤ ≤ −
of M (C) M (C):
2p1p2 ⊕ 2p1p2
Bp+1,,↑r2 0 0 0 Bp−1,,↑p2 r2 0 0 0
BBpp++11,,,,→←rr22 Bp−1,,↑0p2−r2 Bp−1,,↑0p2 r2 00 ,BBpp−−11,,,,←→pp22−−rr22 Bp+01,,↑r2 Bp+01,,↑r2 00  .
(cid:16)Bp+1,,↓r2 Bp−1,,→p2 r2 Bp−1,,←p2−r2 Bp+1,,↑r2 Bp−1,p2−r2 Bp+1,,→r2 Bp+1,,←r2 Bp−1,,↑p2 r2(cid:17)
 − −   − − 
Theorem 6.4. The algebra of Q(r ,r ) with (r ,r ) I is isomorphic to the subalgebra
1 2 1 2
∈
of M (C) M (C) M (C) M (C) given by
4p1p2 ⊕ 4p1p2 ⊕ 4p1p2 ⊕ 4p1p2
T+ 0 0 0 T 0 0 0
r1,r2 p−1 r1,r2
L+ T 0 0 L − T+ 0 0
Rr+1,r2 r−1,p02−r2 T 0 ,R−p1−r1,r2 p1−r01,p2−r2 T+ 0 ,
r1,r2 r−1,p2 r2 p−1 r1,r2 p1 r1,p2 r2
(cid:16)B+ R L − T+  B − R+ L+− − T 
 r1,r2 r−1,p2 r2 −r1,p2 r2 r1,r2  p−1 r1,r2 p1 r1,p2 r2 p1 r1,p2 r2 p−1 r1,r2
 − −   − − − − − − 
T 0 0 0 T+ 0 0 0
r−1,p2 r2 p1 r1,p2 r2
L − T+ 0 0 L+− − T 0 0
R−r1,p2−r2 r01,r2 T+ 0 ,Rp+1−r1,p2−r2 p−1−0r1,r2 T 0 
r−1,p2 r2 r1,r2 p1 r1,p2 r2 p−1 r1,r2
B − R+ L+ T  B+− − R L − T+ (cid:17)
 r−1,p2 r2 r1,r2 r1,r2 r1,p2 r2  p1 r1,p2 r2 p−1 r1,r2 −p1 r1,r2 p1 r1,p2 r2
 − −   − − − − − − 
where
Xα, 0 0 0
r1,↑r2
Xrα1,r2 = XXrrαα11,,,,←→rr22 Xp−1α−0,r↑1,r2 Xp−1α0,r↑1,r2 00 
Xrα1,,↓r2 Xp−1α,r→1,r2 Xp−1α−,r←1,r2 Xrα1,,↑r2
 − − 
and X = T,R,L,B.
By using these matrix realizations, we will get a basis of the space of symmetric linear
functions SLF(g ) as the sums of traces of matrix blocks appeared in the matrix
p1,p2
realizations.
A MATRIX REALIZATION OF g 3
p,q
As it is shown in [11], The space SLF(g ) is isomorphic to the center of g .
p1,p2 p1,p2
This isomorphism is given by c t 1c ⇀ λ where λ is the left integral and t is an
−
7→
invertible element in g such that S2(x) = txt 1 for all x g . Then we can
p1,p2 − ∈ p1,p2
determine the relations between the basis of symmetric linear functions and symmetric
linear functions determined by the action of the central elements and balancing element
of g on integrals.
p,q
In [5], the space of q-characters of g is determined. The most important thing
p1,p2
is that they construct a basis of the space of q-characters. Since the space of the q-
characters is β g β(xy) = β(S2(y)x), x,y g (c.f. [5]) and the square of
{ ∈ p1,p2∗| ∈ p1,p2}
the antipode of g is an inner automorphism by the balancing element g of g (see
p1,p2 p1,p2
[5]), any symmetric linear function is given by β ↼ g where β is a q-character. Under this
correspondence, any element of our basis of SLF(g ) is mapped to a scalar multiple of
p1,p2
an element of the basis of the space of q-characters constructed in [5] (see Appendix A).
This paper is organized as follows. In section 2, we recall the basic definitions and
properties of symmetric linear functions and integrals of Hopf algebras. In section 3, we
recall the definition of the algebra g and its integrals and balancing element given
p1,p2
in [5]. We also introduce the list of non-isomorphic simple modules g given in [5].
p1,p2
In section 4, we construct indecomposable modules as left ideals of g . In section 5,
p1,p2
we find the primitive idempotents of g and show that the indecomposable modules
p1,p2
constructed in section 4 give rise to the list of non-isomorphic indecomposable projective
modules. In addition, we give a decomposition of g into subalgebras. In section 6,
p1,p2
we determine the matrix realization of each subalgebra of g as the subalgebras of
p1,p2
direct sum of matrix algebras. In section 7, we construct the symmetric linear functions
on g which form a basis of the space of symmetric linear functions on g . We also
p1,p2 p1,p2
determine the relations between the basis and the symmetric linear functions given by
the left integral, the balancing element and the central elements. In Appendix, we give a
correspondence between the basis of q-characters of g obtained in [5] and the basis
p1,p2
of symmetric linear functions on g
p1,p2
Acknowledgment
The author is grateful to K. Nagatomo for his continuous encouragement and useful
comments.
2. Preliminaries
In this paper we will always work over the complex number field C. For any vector
space V we denote its dual space HomC(V,C) by V∗.
2.1. Symmetric linear functions. Let A be a finite-dimensional associative algebra.
A symmetric linear function ϕ on A is an element of A which satisfies ϕ(ab) = ϕ(ba)
∗
for all a,b A. We denote the space of symmetric linear functions on A by SLF(A).
∈
If A is a finite-dimensional Hopf algebra, the space SLF(A) coincides with the space of
cocommutative elements of A (see [11]).
∗
4 Y. ARIKE
2.2. Integrals and the square of antipode of Hopf algebras. Let A be a finite-
dimensional Hopf algebra with the coproduct ∆, the counit ε and the antipode S. Any
element of the subspaces
= Λ A aΛ = ε(a)Λ for all a A ,
A
L { ∈ | ∈ }
= Λ A Λa = ε(a)Λ for all a A ,
A
R { ∈ | ∈ }
iscalled a left integral anda right integral ofA, respectively. Since Aisfinite-dimensional,
the space (respectively ) is one-dimensional (cf. [9]). Similarly a left (respectively,
A A
L R
right) integral of the dual Hopf algebra A is an element λ A which satisfies pλ = p(1)λ
∗ ∗
∈
(respectively, λp = p(1)λ) for all p A . Equivalently we can see
∗
∈
= λ A (1 λ)∆(x) = λ(x) for all x A ,
A∗ ∗
L { ∈ | ⊗ ∈ }
= λ A (λ 1)∆(x) = λ(x) for all x A .
A∗ ∗
R { ∈ | ⊗ ∈ }
If = the Hopf algebra A is called unimodular.
A A
L R
Proposition 2.1 ([11]). Let A be a finite-dimensional unimodular Hopf algebra with the
antipode S. Suppose that λ is a left integral of A and that µ is a right integral of A .
∗ ∗
Then
(1) λ(ab) = λ(bS2(a)),
(2) µ(ab) = µ(S2(b)a).
The square of the antipode is called inner if there exists an invertible element t such
that S2(x) = txt 1 for all x A.
−
∈
Denote by ⇀ and ↼ the left and right actions of A on A :
∗
a ⇀ p(b) = p(ba), p ↼ a(b) = p(ab),
for p A and a,b A.
∗
∈ ∈
Proposition 2.2 ([11]). Let A be a finite-dimensional unimodular Hopf algebra with the
antipode S. Suppose that λ is a left integral of A and that µ is a right integral of A . If
∗ ∗
S2 is inner, the linear maps f and f : Z(A) SLF(A) defined by f (c) = t 1c ⇀ λ and
ℓ r ℓ −
→
f (c) = µ ↼ tc are isomorphisms where Z(A) denote the center of A.
r
3. The Hopf algebra g
p,q
3.1. Definition. Let p and p be coprime positive integers. Set q = exp(π√ 1), q =
1 2 2p1−p2 1
q2p2 = exp(π√ 1), and q = q2p1 = exp(π√ 1). We use the q-integers and q-binomial
p1− 2 p2−
coefficients
qn q n m [m] !
− q
[n] = − , = , [n] ! = [n] [2] [1] .
q q q q q
q q 1 n [n] ![m n] ! ···
− − (cid:20) (cid:21)q q − q
We also define
m m m m
[m]1 = [m]q1p2, [m]2 = [m]q2p1, n = n , n = n .
(cid:20) (cid:21)1 (cid:20) (cid:21)q1p2 (cid:20) (cid:21)2 (cid:20) (cid:21)q2p1
Note that
[p m] = ( 1)p2+1[m] and [p m] = ( 1)p1+1[m] .
1 1 1 2 2 2
− − − −
A MATRIX REALIZATION OF g 5
p,q
g is a Hopf algebra over C generated by e ,f and K for i = 1,2 with relations
p1,p2 i i ±
KK 1 = K 1K = 1,
− −
epi = fpi = 0, K2p1p2 = 1, (i = 1,2),
i i
Ke K 1 = q2e , Kf K 1 = q 2f , (i = 1,2),
i − i i i − i− i
e e = e e , f f = f f ,
1 2 2 1 1 2 2 1
Kp2 K p2 Kp1 K p1
− −
[e ,f ] = − , [e ,f ] = − ,
1 1 qp2 q−p2 2 2 qp1 q−p1
1 − 1 2 − 2
as an algebra. The coproduct ∆, counit ε, and antipode S are given by
∆(e ) = e 1+Kp2 e , ∆(e ) = e Kp1 +1 e ,
1 1 1 2 2 2
⊗ ⊗ ⊗ ⊗
∆(f ) = f K p2 +1 f , ∆(f ) = f 1+K p1 f ,
1 1 − 1 2 2 − 2
⊗ ⊗ ⊗ ⊗
∆(K) = K K, ε(e ) = ε(f ) = 0, ε(K) = K,
i i
⊗
S(e ) = K p2e , S(e ) = e K p1,
1 − 1 2 2 −
− −
S(f ) = f Kp2, S(f ) = Kp1f , S(K) = K 1.
1 1 2 2 −
− −
Proposition 3.1. The 2p3p3 elements em1em2fn1fn2Kℓ, where 0 m ,n p 1 and
1 2 1 2 1 2 ≤ i i ≤ i −
0 ℓ 2p p 1, form a basis of g as a vector space over C.
≤ ≤ 1 2 − p1,p2
In [5], it is shown that the restricted quantum groups Uq1p2(sl2) = he1,f1,K1i and
Uq2p1(sl2) = he2,f2,K2i are embedded in gp1,p2 by
e e , f f , K Kp2,
1 1 1 1 1
7→ 7→ 7→
e e , f f , K Kp1
2 2 2 2 2
7→ 7→ 7→
and that
gp1,p2 ∼= Uq1p2(sl2)⊗Uq2p1(sl2)/(K1p2 ⊗1−1⊗K2p2),
where (Kp1 1 1 Kp2) is the Hopf ideal generated by (Kp1 1 1 Kp2).
1 ⊗ − ⊗ 2 1 ⊗ − ⊗ 2
3.2. Integrals and the square of the antipode. The space of left and right integrals
in g and the space of right integrals on g are determined in [5]. The space
p1,p2 p1,p2
spanned by
2p1p2 1
−
ep1−1ep2−1fp1−1fp2−1 Kℓ,
1 2 1 2
ℓ=0
X
coincides with the space of left integrals of g , which also coincides with the space of
p1,p2
right integrals of g . This shows that g is unimodular.
p1,p2 p1,p2
Define the linear functions on g by
p1,p2
(3.2.1) λ(em1em2fn1fn2Kℓ) = δ δ δ δ δ ,
1 2 1 2 m1,p1 1 m2,p2 1 n1,p1 1 n2,p2 1 ℓ,p2 p1
− − − − −
(3.2.2) µ(em1em2fn1fn2Kℓ) = δ δ δ δ δ ,
1 2 1 2 m1,p1 1 m2,p2 1 n1,p1 1 n2,p2 1 ℓ,p1 p2
− − − − −
6 Y. ARIKE
for 0 m p 1 and (p p 1) ℓ p p . Using the induction, we can see that
i i 1 2 1 2
≤ ≤ − − − ≤ ≤
m1 m2 n1 n2
∆(em1em2fn1fn2Kℓ) = qp2(m1−r1)+p2s1(n1−r1)−2p2s1(m1−r1)
1 2 1 2 1
r1=0r2=0s1=0s2=0
X X X X
m m n n
qp1r2(m2−r2)+p1s2(n2−s2)−2p1r2(n2−s2) 1 2 1 2
× 2 r r s s
(cid:20) 1(cid:21)1(cid:20) 2(cid:21)2(cid:20) 1(cid:21)1(cid:20) 2(cid:21)2
er1er2fs1fs2Kp2(m1 r1) p1(n2 s2)+ℓ em1 r1em2 r2fn1 s1fn2 s2Kp1r2 p2s1+ℓ.
× 1 2 1 2 − − − ⊗ 1 − 2 − 1 − 2 − −
Proposition 3.2 ([5]). Each of the spaces of left and right integrals in g is spanned
p1,p2
by λ and µ, respectively.
It is also shown in [5] that the square of the antipode of g is inner and that the
p1,p2
balancing element of g is given as follows.
p1,p2
Proposition 3.3 ([5]). For any x g , we have S2(x) = gxg 1 where g = Kp1 p2.
∈ p1,p2 − −
3.3. Simple modules. The algebra g has 2p p non-isomorphic simple modules (see
p1,p2 1 2
[5]). Let Xα α = ,1 r p ,1 r p be the complete list of non-isomorphic
{ r1,r2 | ± ≤ 1 ≤ 1 ≤ 2 ≤ 2}
simple modules of g . The simple module Xα has a basis formed by weight vectors
p1,p2 r1,r2
bα (r ,r ), 0 n r 1 and 0 n r 1 with the action of g defined by
n1,n2 1 2 ≤ 1 ≤ 1 − ≤ 2 ≤ 2 − p1,p2
Kbα (r ,r ) = αqr1 1 2n1qr2 1 2n2bα (r ,r ),
n1,n2 1 2 1 − − 2 − − n1,n2 1 2
ϕα(n ,r ,r )bα (r ,r ), n = 0,
e1bαn1,n2(r1,r2) = (0,1 1 1 2 n1−1,n2 1 2 n11 6= 0,
ϕα(n ,r ,r )bα (r ,r ), n = 0,
e2bαn1,n2(r1,r2) = (0,2 2 1 2 n1,n2−1 1 2 n22 6= 0,
bα (r ,r ), n = r 1,
f bα (r ,r ) = n1+1,n2 1 2 1 6 1 −
1 n1,n2 1 2 (0, n1 = r1 1,
−
bα (r ,r ), n = r 1,
f bα (r ,r ) = n1,n2+2 1 2 2 6 2 −
2 n1,n2 1 2 (0, n2 = r2 1,
−
where
(3.3.1) ϕα(n ,r ,r ) = αp2( 1)r2 1[n ] [r n ] ,
1 1 1 2 − − 1 1 1 − 1 1
(3.3.2) ϕα(n ,r ,r ) = αp1( 1)r1 1[n ] [r n ] ,
2 2 1 2 − − 2 2 2 − 2 2
for 1 n r 1. We note that
i i
≤ ≤ −
(3.3.3) ϕ α(k ,p r ,r ) = ϕα(k ,p r ,p r ),
−1 1 1 − 1 2 1 1 1 − 1 2 − 2
(3.3.4) ϕα(n ,r ,r ) = ϕ α(n ,r ,p r ),
1 1 1 2 −1 1 1 2 − 2
for 1 n r 1 and 1 k p r 1, and that
1 1 1 1 1
≤ ≤ − ≤ ≤ − −
(3.3.5) ϕ α(k ,r ,p r ) = ϕα(k ,p r ,p r ),
−2 2 1 2 − 2 2 2 1 − 1 2 − 2
(3.3.6) ϕα(n ,r ,r ) = ϕ α(n ,p r ,r ),
2 2 1 2 −2 1 1 − 1 2
for 1 n r 1 and 1 k p r 1.
2 2 2 2 2
≤ ≤ − ≤ ≤ − −
A MATRIX REALIZATION OF g 7
p,q
4. Construction of indecomposable left ideals
In this section we construct left ideals which are isomorphic to Pα whose socle is
r1,r2
isomorphic to a simple module Xα for (r ,r ) = (p ,p ). The structure of module Pα
r1,r2 1 2 6 1 2 r1,r2
is described in [5].
4.1. Highest weight vectors in g . For 1 r p and 1 s r , let us set
p1,p2 ≤ i ≤ i ≤ i ≤ i
2p1p2 1
−
vrα1,r2(s1,s2) = (αq1−(r1−2s1+1)q2−(r2−2s2+1))ℓKℓ,
ℓ=0
X
and
bαn1,↓,n2(r1,r2,s1,s2) = f1n1f2n2ep11−1ep22−1f1p1−s1f2p2−s2vrα1,r2(s1,s2),
for 0 n r 1. We then see that
i i
≤ ≤ −
(4.1.1) Kbα, (r ,r ,s ,s ) = αqr1 1 2n1qr2 1 2n2bα, (r ,r ,s ,s )
n1↓,n2 1 2 1 2 1 − − 2 − − n1↓,n2 1 2 1 2
for 0 n r 1 and 0 n r 1. We also have
1 1 2 2
≤ ≤ − ≤ ≤ −
(4.1.2) f1bαn1,↓,n2(r1,r2,s1,s2) = bαn1,↓+1,n2(r1,r2,s1,s2),
(4.1.3) f2bαn1,↓,n2(r1,r2,s1,s2) = bαn1,↓,n2+1(r1,r2,s1,s2)
for 0 n r 2 and 0 n r 2.
1 1 2 2
≤ ≤ − ≤ ≤ −
Lemma 4.1 ([6]). For 1 n p 1, i,j 1,2 and i = j, the following relations
i i
≤ ≤ − ∈ { } 6
hold in g :
p1,p2
[ei,fini] = [ni]ifini−1qi−pj(ni−1)Kqppjj −qq−ippjj(ni−1)K−pj
i − i
= [ni]iqipj(ni−1)Kqppjj−qqi−−ppjj(ni−1)K−pjfini−1,
i − i
[enii,fi] = [ni]ienii−1qipj(ni−1)Kqppjj−qqi−−ppjj(ni−1)K−pj
i − i
= [ni]iqi−pj(ni−1)Kqppjj −qq−ippjj(ni−1)K−pjeini−1.
i − i
Lemma 4.2. We have
(4.1.4) e bα, (r ,r ,s ,s )
1 n1↓,n2 1 2 1 2
= ϕα1(n1,r1,r2)bαn1,↓ 1,n2(r1,r2,s1,s2), 1 ≤ n1 ≤ r1 −1,
−
(0, n1 = 0,
(4.1.5) e bα, (r ,r ,s ,s )
2 n1↓,n2 1 2 1 2
= ϕα2(n2,r1,r2)bαn1,↓,n2−1(r1,r2,s1,s2), 1 ≤ n2 ≤ r2 −1,
(0, n2 = 0,
8 Y. ARIKE
Proof. By Lemma 4.1, we obtain
e bα, (r ,r ,s ,s )
1 n1↓,n2 1 2 1 2
=e1f1n1f2n2ep11−1ep22−1f1p1−s1f2p2−s2vrα1,r2(s1,s2)
=[n1]1q1p2(n1−1)Kqp1p22−−qq1−1−pp22(n1−1)K−p2bαn1,↓−1,n2(r1,r2,s1,s2)
=αp2(−1)r2−1[n1]1[r1 −n1]bαn1,↓−1,n2(r1,r2,s1,s2),
for 1 n r 1. Similarly we have (4.1.5). (cid:3)
1 1
≤ ≤ −
The space spanned by bα (r ,r ,s ,s ) with 0 n r 1 and 0 n r 1 is
n1,n2 1 2 1 2 ≤ 1 ≤ 1 − ≤ 2 ≤ 2 −
isomorphic to the simple module Xα . This fact will be proved in next subsection.
r1,r2
Proposition 4.3. (1) For 1 r p 1,
1 1
≤ ≤ −
α,
b0,n↓2(r1,r2,s1,s2)
p1 r1
−
=f1f2n2 γmα1(r1,r2)ep11−m1ep22−1f1p1−s1−m1f2p2−s2vrα1,r2(s1,s2),
m1=1
X
where
p1 r1 1
− −
γα (r ,r ) = ϕ α(k ,p r ,r ).
m1 1 2 −1 1 1 − 1 2
k1=p1−Yr1−(m1−1)
(2) For 1 r p 1,
2 2
≤ ≤ −
α,
bn1↓,0(r1,r2,s1,s2)
p2 r2
−
=f1n1f2 δmα2(r1,r2)ep11−1ep22−m2f1p1−s1f2p2−s2−m2vrα1,r2(s1,s2),
m2=1
X
where
p2 r2 1
− −
δα (r ,r ) = ϕ α(k ,r ,p r ).
m2 1 2 −2 2 1 2 − 2
k2=p2−Yr2−(m2−1)
(3) For 1 r p 1 and 1 r p 1,
1 1 2 2
≤ ≤ − ≤ ≤ −
p1 r1p2 r2
− −
bα0,,0↓(r1,r2,s1,s2) =f1f2 γmα1(r1,r2)δmα2(r1,r2)
m1=1m2=1
X X
×e1p1−m1ep22−m2f1p1−s1−m1f2p2−s2−m2vrα1,r2(s1,s2).
Proof. We only prove (1). proofs of others are similar.
A MATRIX REALIZATION OF g 9
p,q
By Lemma 4.1, we have
f1e1p1−m1ep22−1f1p1−s1−m1f2p2−s2vrα1,r2(s1,s2)
=ep11−m1ep22−1f1p1−s1−m1+1f2p2−s2vrα1,r2(s1,s2)
−[ep11−m1,f1]ep22−1f1p1−s1−m1f2p2−s2vrα1,r2(s1,s2)
=ep11−m1ep22−1f1p1−s1−m1+1f2p2−s2vrα1,r2(s1,s2)
−ϕ−1α(m1,p1 −r1,r2)ep11−1−m1ep22−1f1p1−s1−m1f2p2−s2vrα1,r2(s1,s2).
Thus we have
p1 r1
−
f1 γmα1(r1,r2)ep11−m1ep22−1f1p1−s1−m1f2p2−s2vrα1,r2(s1,s2)
m1=1
X
p1 r1
−
= γmα1(r1,r2)e1p1−m1ep22−1f1p1−s1−m1+1f2p2−s2vrα1,r2(s1,s2)
m1=1
X
p1 r1 1
− −
− γmα1+1(r1,r2)ep11−m1−1ep22−1f1p1−s1−m1f2p2−s2vrα1,r2(s1,s2)
m1=1
X
= ep11−1ep22−1f1p1−s1f2p2−s2vrα1,r2(s1,s2).
(cid:3)
Set
r1 1 r2 1
− −
Φα(r ,r ) = 2p p ϕα(i ,r ,r ) ϕα(k ,r ,r )
1 2 1 2 1 1 1 2 2 2 1 2
i1=1 i2=1
Y Y
p1 r1 1 p2 r2 1
− − − −
ϕ α(k ,p r ,r ) ϕ α(k ,r ,p r ),
× −1 1 1 − 1 2 −2 2 1 2 − 2
kY1=1 kY2=1
r1−1 1 p1−r1−1 1
Ψα(r ,r ) = + ,
1 1 2 ϕα(i ,r ,r ) ϕ α(j ,p r ,r )
i1=1 1 1 1 2 j1=1 −1 1 1 − 1 2
X X
r2−1 1 p2−r2−1 1
Ψα(r ,r ) = + .
2 1 2 ϕα(i ,r ,r ) ϕ α(j ,r ,p r )
i2=1 2 2 1 2 j2=1 −2 2 1 2 − 2
X X
Then we can see that Φα(r ,r ) = Φ α(p r ,r ) = Φ α(r ,p r ) = Φα(p r ,p r )
1 2 − 1 1 2 − 1 2 2 1 1 2 2
− − − −
and that Ψα(r ,r ) = Ψ α(p r ,r ) = Ψ α(r ,p r ) = Ψα(p r ,p r ) for i = 1,2
i 1 2 −i 1− 1 2 −i 1 2− 2 i 1− 1 2− 2
by (3.3.3)-(3.3.6). Now we set
1
Bα, (r ,r ,s ,s ) = bα, (r ,r ,s ,s ).
n1↓,n2 1 2 1 2 Φα(r ,r ) n1↓,n2 1 2 1 2
1 2
4.2. Indecomposable left ideals Pα (s ,s ) and Pα (s ,s ). Let us first fix 1
r1,p2 1 2 p1,r2 1 2 ≤
r p 1 and r = p . We set
1 1 2 2
≤ −
α,
(4.2.1) Bk1←,n2(r1,p2,s1,s2)
=e1p1−r1−1−k1f2n2 pm11−=r11γmα1(r1,p2)ep11−m1ep22−1f1p1−s1−m1f2p2−s2vrα1,p2(s1,s2),
PΦα(r1,p2) pj11=−kr11+−11ϕ−1α(j1,p1 −r1,p2)
Q
10 Y. ARIKE
(4.2.2) Bα, (r ,p ,s ,s )
n1↑,n2 1 2 1 2
fn1fn2 p1−r1
=Φα1(r ,2p ) γmα1(r1,p2)ep11−1−m1ep22−1f1p1−s1−m1f2p2−s2vrα1,r2(r1,r2)
1 2
m1=1
X
Ψα(r ,p )Bα (r ,p ,s ,s ),
− 1 1 2 n1,n2 1 2 1 2
(4.2.3) Bkα1,→,n2(r1,p2,s1,s2) = f1r1+k1B0α,,n↑2(r1,p2,s1,s2),
for 0 n r 1, 0 k p r 1 and 0 n p 1.
1 1 1 1 1 2 2
≤ ≤ − ≤ ≤ − − ≤ ≤ −
Lemma 4.4. For 0 n p 1, the following relations hold:
2 2
≤ ≤ −
(4.2.4)
KBkα1,←,n2(r1,p2,s1,s2) = −αq1p1−r1−1−2k1q2r2−1−2n2Bkα1,←,n2(r1,p2,s1,s2),
(4.2.5)
KBα, (r ,p ,s ,s ) = αqr1 1 2n1qr2 1 2n2Bα, (r ,p ,s ,s ),
n1↑,n2 1 2 1 2 1 − − 2 − − n1↑,n2 1 2 1 2
(4.2.6)
KBkα1,→,n2(r1,p2,s1,s2) = −αq1p1−r1−1−2k1q2r2−1−2n2Bkα1,→,n2(r1,p2,s1,s2),
(4.2.7)
α,
e1Bk1←,n2(r1,p2,s1,s2)
= ϕ−1α(k1,p1 −r1,p2)Bkα1,←1,n2(r1,p2,s1,s2), 1 ≤ k1 ≤ p1 −r1 −1,
−
(0, k1 = 0,
(4.2.8)
α,
f1Bk1←,n2(r1,p2,s1,s2)
α,
= Bk1←+1,n2(r1,p2,s1,s2), 0 ≤ k1 ≤ p1 −r1 −2,
α,
(B0,n↓2(r1,p2,s1,s2), k1 = p1 −r1 −1,
(4.2.9)
α,
f Bα, (r ,p ,s ,s ) = Bn1↓+1,n2(r1,p2,s1,s2), 0 ≤ n1 ≤ r1 −2,
1 n1↓,n2 1 2 1 2 (0, n1 = r1 1,
−
(4.2.10)
e Bα, (r ,p ,s ,s )
1 n1↑,n2 1 2 1 2
= ϕα1(n1,r1,p2)Bnα1,↑ 1,n2(r1,p2,s1,s2)+Bnα1,↓ 1,n2(r1,p2,s1,s2), 1 ≤ n1 ≤ r1 −1,
− −
α,
(Bp1↑r1 1,n2(r1,p2,s1,s2), n1 = 0,
− −
(4.2.11)
α,
f1Bn1↑ 1,n2(r1,p2,s1,s2)
−
α,
= Bn1↑+1,n2(r1,p2,s1,s2), 0 ≤ n1 ≤ r1 −2,
α,
(B0,n→2(r1,p2,s1,s2), n1 = r1 −1,