Highest weight modules over quantum queer Lie superalgebra U_q(q(n))

0.39 MB·

by  Dimitar Grantcharov

HIGHEST WEIGHT MODULES OVER
QUANTUM QUEER SUPERALGEBRA U (q(n))
q
9
0 DIMITARGRANTCHAROV1, JI HYEJUNG2,3, SEOK-JINKANG2 ANDMYUNGHO KIM2,3
0
2
Abstract. In this paper, we investigate the structure of highest weight modules over the
c
e quantumqueersuperalgebraUq(q(n)). Thekeyingredientsarethetriangulardecomposition
D
of Uq(q(n)) and the classification of finite dimensional irreducible modules over quantum
4 Clifford superalgebras. The main results we prove are the classical limit theorem and the
2 complete reducibility theorem for Uq(q(n))-modules in the category Oq≥0.

]
T
R
Introduction
.
h
t
a Since its inception, the representation theory of Lie superalgebras has been known to
m
be much more complicated than the corresponding theory of Lie algebras. One of the Lie
[
superalgebra series attracts special attention due to its resemblance of the Lie algebra gln
2
on the one hand and because of the unique properties of its structure and representations
v
5 on the other. This is the so-called queer (or strange) Lie superalgebra q(n) which consists
6
of all endomorphisms of Cnn that commute with an odd automorphism P of Cnn such that
2 | |
0 P2 = Id. The queer nature of q(n) is partly due to the nonabelian structure of its Cartan
.
6 subsuperalgebra h having a nontrivial odd part h . Another unique property of q(n) is that,
¯1
0
although it has no invariant bilinear form, it admits an invariant odd bilinear form. Because
9
0 ofthenonabelianstructureofh,thestudyofthehighestweightmodulesofq(n)requiressome
:
v tools in addition to the standard technique. For example, the highest weight space v of an
λ
i
X irreducible highest weight q(n)-module V(λ) has a Clifford module structure. The case when
r V(λ) is a tensor module; i.e., a submodule of some tensor power V r of the natural q(n)-
a ⊗
module V = Cnn, was treated first by Sergeev in 1984. In [Se2] Sergeev established several
|
important results, among which are the complete reducibility of V r, a character formula of
⊗
V(λ), andananalogofthefundamentalSchur-Weyl duality, often referredasSergeev duality.
The characters of all simple finite-dimensional q(n)-modules have been found by Penkov and
Serganovain1996(see[PS2]and[PS3])viaanalgorithmusingasupergeometricversionofthe
Borel-Weil-Bott Theorem. In 2004 Brundan, [B], obtained the character formula of Penkov
1This research was supported bya UT Arlington REPGrant.
2This research was supported byKRFGrant # 2007-341-C00001.
3This research was supported byBK21 Mathematical Sciences Division.
1
2 GRANTCHAROV,JUNG,KANG,KIM
and Serganova using a different approach and formulated a conjecture for the characters of
the irreducible modules in the category . Important results related to the simplicity of the
O
highest weight q(n)-modules were obtained recently by Gorelik in [G].
In this paper we initiate the study of highest weight representations of the quantum su-
peralgebra U (q(n)). The aim of this paper is twofold. We want to study highest weight
q
U (q(n))-modules on the one hand, and to build the foundations of the crystal bases theory
q
for the tensor modules of U (q(n)) on the other. The latter problem will be treated in a
q
future work.
A quantum deformation of the universal enveloping algebra of q(n) was constructed first
byOlshanskiin[O]. Olshanski’sconstruction isaflatdeformation oftheuniversalenveloping
algebra U(q(n)) of q(n) and is a quantum enveloping superalgebra in the sense of Drinfeld
([Dr], 7). The idea in [O] is to apply a suitable modification of the procedure used by
§
Faddeev, Reshetikhin, and Takhtajan in [RTF] – using an element S in End(Cnn) 2 that
| ⊗
satisfies the quantum Yang-Baxter equation. However, as pointed out by Olshanski, the r-
matrix r q(n) 2 does not satisfy the classical Yang-Baxter equation. Thus no quantum
⊗
∈
analogue of U(q(n)) can be a quasi-triangular Hopf algebra.
In the present paper, based on the description of Olshanski, we give a presentation of
U (q(n)) in terms of generators and relations so that the relations are quantum deformations
q
of the relations of q(n) obtained in [LS]. Using this presentation, we find a natural triangular
decomposition of U (q(n)), and then introduce the notion of highest weight modules and
q
Weyl modules. Similarly to the case of q(n), in order to study highest weight modules, one
has to describe the modules over the quantum Clifford superalgebra Cliff (λ) for a weight λ
q
of q(n). These modules, as we show in Section 3, do not have the same structure as the ones
over the classical Clifford superalgebra Cliff(λ). For example, the irreducible modules over
Cliff (λ) are parity invariant for much larger set of weights λ, compared with the irreducibles
q
over Cliff(λ).
In the last two sections of the paper we focus on the category 0 of finite dimensional
Oq≥
U (q(n))-modules all whose weights are of the form λ ǫ + + λ ǫ (λ Z ). One of
q 1 1 n n i 0
··· ∈ ≥
our main results is a classical limit theorem for the irreducible modules in 0. Due to the
Oq≥
structure of the quantum Clifford superalgebra, the classical limit theorem is non-standard,
asitisnottrueingeneralthattheclassicallimitV1 ofanirreduciblehighestweightU (q(n))-
q
module Vq(λ) is V(λ). In fact, as we show in Section 5, if λ has even number of nonzero
coordinates λ > ... > λ , then chV1 = 2chV(λ). The “queer” version of the classical limit
1 2k
theorems are Theorem 5.14 and Theorem 5.16. With the aid of the classical limit theorems
we obtain another important result in the last section: the category 0 is semisimple.
Oq≥
HIGHEST WEIGHT MODULES OVER THE QUANTUM QUEER SUPERALGEBRA Uq(q(n)) 3
The organization of the paper is as follows. In Section 1 we recall some definitions and
basic results about q(n). The realization of U (q(n)) and its triangular decomposition is
q
provided in Section 2. Section 3 is devoted to the study of the quantum Clifford superalgebra
and its modules. In Section 4 we introduce the notion of highest weight modules and Weyl
modules. In particular, we show that every Weyl module Wq(λ) has a unique irreducible
quotient Vq(λ). The classical limit theorem for the category 0 is proved in Section 5 and
Oq≥
the complete reducibility of U (q(n))-modules in 0 is established in the last section.
q Oq≥
1. The Lie superalgebra q(n) and its representations
The ground field in this section will be C. By Z and Z we denote the nonnegative
0 >0
≥
integers and strictly positive integers, respectively. We set Z = Z/2Z. Every vector space
2
V = V¯0 V¯1 over C is Z2-graded with even part V¯0 and odd part V¯1. We will write dimV =
⊕
m n if dimV = m and dimV = n. By Π we denote the parity change functor; i.e., ΠV is a
¯0 ¯1
|
vector space for which ΠV = V and ΠV = V . The direct sum of r copies of a vector space
¯0 ¯1 ¯1 ¯0
V will be written as V r.
⊕
The Lie subsuperalgebra g = q(n) of gl(n n) is defined in matrix form by
|
A B
g =q(n) := A,B gl .
n
( B A ! ∈ )
(cid:12)
(cid:12)
(cid:12)
By definition, a subsupealgebra h = h h of g is a Cartan subsuperalgebra, if it is a self-
¯0 ¯1
⊕
normalizingnilpotentsubsuperalgebra. Everysuchhhasanontrivialoddparth . Wefixhto
¯1
be the standard Cartan subsuperalgebra, namely the one for which h has a basis k ,...,k
¯0 1 n
{ }
E 0 0 E
i,i i,i
and h has a basis k ,...,k , where k := , k := and E is
¯1 { ¯1 n¯} i 0 Ei,i ! ¯i Ei,i 0 ! i,j
the n n matrix having 1 in the (i,j) position and 0 elsewhere. One should note that all
×
Cartan subsuperalgebras of g are conjugate to h. Let ǫ ,...,ǫ be the basis of h dual to
{ 1 n} ∗¯0
k ,...,k . Wedenotek k byh fori= 1,2, ,n 1. Therootsystem∆ = ∆ ∆ ofg
1 n i i+1 i ¯0 ¯1
{ } − ··· − ∪
hasidentical even andoddparts. Namely, ∆ = ∆ = ǫ ǫ 1< i = j < n . Inparticular,
¯0 ¯1 i j
{ − | 6 }
the root space decomposition g = g has the property that g has dimension 11 for
α ∆ α α |
everyα ∈ ∆. Setαi := ǫi−ǫi+1. LetLQ =∈ in=−11ZαibetherootlatticeandQ+ = in=−11Z≥0αi
be the positive root lattice. The notation Q = Q will also be used. There is a partial
L − − + P
ordering on h defined by λ µ if and only if λ µ Q for λ,µ h . The root space g
∗¯0 ≥ − ∈ + ∈ ∗¯0 αi
E 0 0 E
i,i+1 i,i+1
is spanned by e := and e := , while g is spanned
i 0 Ei,i+1 ! ¯i Ei,i+1 0 ! −αi
4 GRANTCHAROV,JUNG,KANG,KIM
E 0 0 E
by fi := i+01,i Ei+1,i ! and f¯i := Ei+1,i i+01,i !. Let P := ni=1Zǫi be the weight
lattice of g and denote by P := n Zk the dual weight lattice. L
∨ i=1 i
Let I := 1,2, ,n 1 and J := 1,2, ,n .
{ ··· − } L { ··· }
Proposition 1.1. [LS]TheLie superalgebra gisgenerated bythe elementse ,e ,f ,f (i I),
i ¯i i ¯i ∈
h and k (l J) with the following defining relations:
¯0 ¯l ∈
[h,h] = 0 for h,h h ,
′ ′ ¯0
∈
[h,e ] = α (h)e , [h,e ] = α (h)e for h h , i I,
i i i ¯i i ¯i ∈ ¯0 ∈
[h,f ]= α (h)f , [h,f ] = α (h)f for h h , i I,
i − i i ¯i − i ¯i ∈ ¯0 ∈
[h,k ] = 0 for h h , l J,
¯l ∈ ¯0 ∈
[e ,f ]= δ (k k ), [e ,f ]= δ (k k ) for i,j I,
i j ij i− i+1 i ¯j ij ¯i− i+1 ∈
[e ,f ]= δ (k k ), [k ,e ]= α (k )e for i,j I, l J,
¯i j ij ¯i− i+1 l i i l i ∈ ∈
[k ,f ] = α (k )f , [e ,f ]= δ (k +k ) for i,j I, l J,
l i − i l i ¯i ¯j ij i i+1 ∈ ∈
e if l = i,i+1
i
[k ,e ] = for i I, l J,
¯l ¯i 0 otherwise ∈ ∈

f if l = i,i+1
i
[k ,f ] = for i I, l J,
¯l ¯i 0 otherwise ∈ ∈

[e ,e ]= [e ,e ] = [f ,f ] = [f ,f ]= 0 for i,j I, i j = 1,
i ¯j ¯i ¯j i ¯j ¯i ¯j ∈ | − | 6
[e ,e ]= [f ,f ]= 0 for i,j I, i j > 1,
i j i j
∈ | − |
[e ,e ]= [e ,e ],[e ,e ] = [e ,e ],
i i+1 ¯i i+1 i i+1 ¯i i+1
[f ,f ]= [f ,f ],[f ,f ]= [f ,f ],
i+1 i i+1 ¯i i+1 ¯i i+1 i
[k ,k ]= δ 2k for i,j J,
¯i ¯j ij i ∈
[e ,[e ,e ]] = [e ,[e ,e ]] = 0 for i,j I, i j = 1,
i i j ¯i i j ∈ | − |
[f ,[f ,f ]] = [f ,[f ,f ]] = 0 for i,j I, i j = 1.
i i j ¯i i j ∈ | − |
Remark. We modified the relations given in [LS]. More precisely, we replaced the relations
[e ,[e ,e ]] = 0 for i,j I, i j = 1,
(1.1) ¯i i ¯j ∈ | − |
[f ,[f ,f ]] = 0 for i,j I, i j = 1
¯i i ¯j ∈ | − |
HIGHEST WEIGHT MODULES OVER THE QUANTUM QUEER SUPERALGEBRA Uq(q(n)) 5
by
[e ,e ] = [e ,e ],[e ,e ]= [e ,e ],
i i+1 ¯i i+1 i i+1 ¯i i+1
(1.2)
[f ,f ]= [f ,f ],[f ,f ] = [f ,f ].
i+1 i i+1 ¯i i+1 ¯i i+1 i
Since (1.1) can be derived from (1.2) (and other ones), we can easily see that these two
presentations are equivalent.
TheuniversalenvelopingalgebraU(g)isobtainedfromthetensoralgebraT(g)byfactoring
out by the ideal generated by the elements [u,v] u v+( 1)αβv u, where α,β Z , u
2
− ⊗ − ⊗ ∈ ∈
g , v g . Let U+ (respectively, U0 and U ) be the subalgebra of U(g) generated by the
α β −
∈
elements e ,e (i I) (respectively, by k ,k (i J) and by f ,f (i I)). By the Poincar´e-
i ¯i ∈ i ¯i ∈ i ¯i ∈
Birkhoff-Witt theorem, the universal enveloping algebra has the triangular decomposition:
(1.3) U(g) ∼= U− U0 U+.
⊗ ⊗
A g-module V is called a weight module if it admits a weight space decomposition
V = V , where V = v V hv = µ(h)v for all h h .
µ µ ¯0
{ ∈ | ∈ }
µ h∗
M∈ ¯0
For a weight g-module M denote by wt(M) the set of weights λ h for which M = 0.
∈ ∗¯0 λ 6
Every submodule of a weight module is also a weight module. If dimCVµ < ∞ for all µ ∈ h∗¯0,
the character of V is defined to be
chV = (dimCVµ) eµ,
µ h∗
X∈ ¯0
where eµ are formal basis elements of the group algebra C[h ] with the multiplication given
∗¯0
by eλeµ = eλ+µ for all λ,µ h .
∈ ∗¯0
Denote by b the standard Borel subsuperalgebra of g generated by k ,k (l J) and e , e
+ l ¯l ∈ i ¯i
(i I). A weight module V is called a highest weight module if it is generated over g by a
∈
finite dimensional irreducible b -submodule (see [PS1, Definition 4]).
+
Proposition 1.2. [P] Let v be a finite dimensional irreducible Z -graded b -module.
2 +
(1) The maximal nilpotent subsuperalgebra n of b acts on v trivially.
+
(2) For any weight µ h , consider the symmetric bilinear form F (u,v) := µ([u,v]) on
∈ ∗¯0 µ
h and let Cliff(µ) be the Clifford superalgebra of the quadratic space (h ,F ). Then
¯1 ¯1 µ
there exists a unique weight λ h such that v is endowed with a canonical Z -graded
∈ ∗¯0 2
Cliff(λ)-module structure and v is determined by λ up to Π.
(3) h acts on v by the weight λ determined in (2).
¯0
6 GRANTCHAROV,JUNG,KANG,KIM
From the above proposition, we know that the dimension of the highest weight space of a
highest weight g-module with highest weight λ is the same as the dimension of an irreducible
Cliff(λ)-module. Ontheother handall irreducibleCliff(λ)-modules have thesame dimension
(see, forexample, [ABS,Table2]). Thusthedimensionofthehighestweightspaceisconstant
for all highest weight modules with highest weight λ.
Definition 1.3. Let v(λ) be the irreducible b -module determined by λ up to Π. The Weyl
+
module W(λ) of g with highest weight λ is defined to be
W(λ):= U(g) v(λ).
⊗U(b+)
Note that the structure of W(λ) is determined by λ up to Π.
Remark. One may define the Verma module corresponding to λ by M(λ) := U(g)
⊗U(b+)
Cliff(λ). SincetheVermamodulesarenothighestweightmodules,theywillnotbeconsidered
in this paper.
We will denote by Λ+ and Λ+ the set of gl -dominant integral weights and the set of
¯0 n
g-dominant integral weights, respectively. These are given by
Λ+ := λ ǫ + +λ ǫ h λ λ Z for all i I
¯0 { 1 1 ··· n n ∈ ∗¯0 | i− i+1 ∈ ≥0 ∈ }
Λ+ := λ ǫ + +λ ǫ Λ+ λ = λ λ = λ = 0 for all i I .
{ 1 1 ··· n n ∈ ¯0 | i i+1 ⇒ i i+1 ∈ }
Proposition 1.4. [P]
(1) For any weight λ, W(λ) has a unique maximal submodule N(λ).
(2) For each finite dimensional irreducible g-module V, there exists a unique weight λ
∈
Λ+ such that V is a homomorphic image of W(λ).
¯0
(3) V(λ) := W(λ)/N(λ) is finite dimensional if and only if λ Λ+.
∈
Now we restrict our attention to the following subcategory of the category of finite dimen-
sional g-modules.
Definition 1.5. Set P := λ = λ ǫ +...+λ ǫ P λ 0 for all j = 1, ,n . The
0 1 1 n n j
≥ { ∈ | ≥ ··· }
category 0 consistsoffinitedimensionalU(g)-modulesM withweight spacedecomposition
≥
O
M = M such that wt(M) P .
λ∈P λ ⊂ ≥0
CleLarly, 0 is closed under finite direct sum, tensor product and taking submodules and
≥
O
quotient modules. Because a q(n)-module in 0 can be decomposed into a direct sum of
≥
O
irreducible highest weight gl -modules, one can easily prove the following proposition (see,
n
for example, [HK, Theorem 7.2.3]).
HIGHEST WEIGHT MODULES OVER THE QUANTUM QUEER SUPERALGEBRA Uq(q(n)) 7
Proposition 1.6. For each λ Λ+ P , V(λ) is an irreducible U(g)-module in the category
0
∈ ∩ ≥
0. Conversely, every irreducible U(g)-module in the category 0 has the form V(λ) for
≥ ≥
O O
some λ Λ+ P .
0
∈ ∩ ≥
In [Se1], Sergeev has presented an explicit set of generators of Z = (U(g)), the center
Z
of U(g), and showed that each Weyl module W(λ) (λ h ) admits a central character.
∈ ∗¯0
Let χλ HomC(Z,C) be the central character afforded by W(λ); i.e., every element z Z
∈ ∈
acts on W(λ) as scalar multiplication by χ (z). Following [B, (2.12)], to each weight λ =
λ
λ ǫ + +λ ǫ P, one can assign a formal symbol
1 1 n n
··· ∈
δ(λ) := δ + +δ
λ1 ··· λn
such that δ = 0 and δ = δ .
0 i i
− −
Proposition 1.7. [B, Theorem 4.19], [PS2, Proposition 1.1] For λ,µ P, χ = χ if and
λ µ
∈
only if δ(λ) = δ(µ).
The following proposition will be very useful in Section 5.
Proposition 1.8. Let V be a finite dimensional highest weight module over g with highest
weight λ Λ+ P . Then V is isomorphic to an irreducible highest weight module V(λ).
0
∈ ∩ ≥
Proof. If V is reducible, since it is finite dimensional, it contains a nonzero proper irreducible
submoduleW. Then W is isomorphic to an irreducible highest weight module V(µ) for some
weight µ Λ+ P by Proposition 1.4. We know that µ (cid:0) λ and χ = χ . But, by
0 λ µ
∈ ∩ ≥
Proposition 1.7, δ(λ) = δ(µ). Since λ,µ Λ+ P , we have λ = µ, which is a contradiction.
0
∈ ∩ ≥
ThusV is irreducibleandby Proposition 1.4, itmustbeisomorphicto theirreduciblehighest
weight module V(λ) up to Π. (cid:3)
The next proposition gives a sufficient condition for the finite dimensionality of a highest
weight g-module.
Proposition 1.9. Let V be a highest weight module over g with highest weight λ Λ+. If
∈
fλ(hi)+1v = 0 for all v V and i I, then V is finite dimensional.
i ∈ λ ∈
Proof. Let x ,x ,...,x and y ,y ,...,y be bases of g and g , respectively. Then
1 2 r 1 2 r ¯0 ¯1
{ } { }
by the Poincar´e-Birkhoff-Witt theorem, U(g) has a basis consisting of elements of the form
yǫ1yǫ2 yǫrxn1xn2 xnr where ǫ = 0 or 1 and n N 0 . Because yǫ1yǫ2 yǫr ǫ =
1 2 ··· r 1 2 ··· r j j ∈ ∪{ } { 1 2 ··· r | j
0,1 is a finite set, it is enough to show that U(g )V is finite dimensional. For any v V ,
¯0 λ λ
} ∈
we know that U(g )v is a highest weight module over g with highest weight λ satisfying
¯0 ¯0
fλ(hi)+1v = 0 for all i I. Thus it is finite dimensional. Since U(g )V U(g )v, we
i ∈ ¯0 λ ⊂ ¯0
have the desired result. vX∈Vλ (cid:3)
8 GRANTCHAROV,JUNG,KANG,KIM
We say that a weight λ = λ ǫ + +λ ǫ h is α-typical if α= ǫ ǫ and λ +λ = 0.
1 1 ··· n n ∈ ∗¯0 i− j i j 6
In [Se2], Sergeev proved the following character formula for V(λ) (λ Λ+ P ):
0
∈ ∩ ≥
dimv
(1.4) chV(λ) = λ sgn w w eλ+ρ0 (1+e α) ,
−
D
wX∈W (cid:16) α∈Y∆+¯0, (cid:17)
λ is α tyipical
−
wherev is anirreducibleCliff(λ)-module, W is theWeyl groupof g = gl , ρ = 1 α
λ ¯0 n 0 2 α ∆+
∈ ¯0
and D = w W sgn w ew(ρ0) is the Weyl denominator. In [PS2], the formula (1.4)Pis called
∈
the generic character formula and an explicit algorithm for computing the character of an
P
arbitrary finite dimensional irreducible g-module is presented.
2. The quantum superalgebra U (q(n))
q
In [O], Olshanski constructed the quantum deformation U (q(n)) of the universal envelop-
q
ing algebra of q(n). The quantum superalgebra U (q(n)) is defined to be the associative
q
algebra over C(q) generated by L , i j, with defining relations
ij
≤
L L = L L = 1,
ii i, i i, i ii
− − − −
( 1)p(i,j)p(k,l)qϕ(j,l)L L + k j <l θ(i,j,k)(q q 1)L L
ij kl − il kj
− { ≤ } −
(2.1) + i l < j k θ( i, j,k)(q q−1)Li, lLk, j
{ ≤ − ≤ − } − − − − −
=qϕ(i,k)L L + k < i l θ(i,j,k)(q q 1)L L
kl ij − il kj
{ ≤ } −
+ l i < k j θ( i, j,k)(q q 1)L L ,
− i,l k,j
{− ≤ − ≤ } − − − − −
0 if ij > 0
whereϕ(i,j) = δ sgn(j), θ(i,j,k) = sgn(sgn(i)+sgn(j)+sgn(k)),p(i,j) =
|i|,|j| 1 if ij < 0,

for any indices i j, k l in 1, n and the symbol (the dots stand for some
≤ ≤ {± ··· ± } {···} 
inequalities) is equal to 1 if all of these inequalities are fulfilled and 0 otherwise.
Following [O,Remark7.3], weconsiderthesetofgeneratorsofU (g) = U (q(n))asfollows:
q q
1 1
qki := L , q ki := L , e := L , f := L ,
(2.2) i,i − −i,−i i −q−q−1 −i−1,−i i q−q−1 i,i+1
1 1 1
e := L , f := L , k := L .
¯i −q q−1 −i−1,i ¯i −q q−1 −i,i+1 ¯i −q q−1 −i,i
− − −
Our first main result is the following presentation of U (g).
q
Theorem 2.1. The quantum superalgebra U (g) is isomorphic to the unital associative alge-
q
bra over C(q) generated by the elements ei,fi,e¯i,f¯i (i = 1,...,n−1), k¯l (l = 1,...,n), and qh
(h P ), satisfying the following relations
∨
∈
q0 = 1,qh1+h2 = qh1qh2 for h ,h P ,
1 2 ∨
∈
HIGHEST WEIGHT MODULES OVER THE QUANTUM QUEER SUPERALGEBRA Uq(q(n)) 9
qhe q h = qαi(h)e ,qhf q h = q αi(h)f for h P
i − i i − − i ∨
∈
qhk¯iq−h = k¯i,qhe¯iq−h = qαi(h)e¯i,qhf¯iq−h = q−αi(h)f¯i for h∈ P∨
1
eifi−fiei = q q 1 qki−ki+1 −q−ki+ki+1 ,
−
− (cid:16) (cid:17)
qe f f e = e f qf e = e f f e = 0 if i j > 1,
i+1 i i i+1 i i+1 i+1 i i j j i
− − − | − |
eif¯i−f¯iei = q−ki+1k¯i−ki+1q−ki,
qe f f e = e f qf e = e f f e = 0 if i j > 1,
i+1 ¯i− ¯i i+1 i i+1− i+1 i i ¯j − ¯j i | − |
e¯ifi−fie¯i = qki+1k¯i−ki+1qki,
qe f f e = e f qf e = e f f e = 0 if i j > 1,
i+1 i− i i+1 ¯i i+1− i+1 ¯i ¯i j − j ¯i | − |
k¯iei−qeik¯i = e¯iq−ki, qk¯iei−1−ei−1k¯i = −q−kiei−1,
k e e k = 0 for j = i and j = i 1,
¯i j − j ¯i 6 6 −
k¯ifi−qfik¯i = −f¯iqki, qk¯ifi−1−fi−1k¯i = qkifi−1,
k f f k = 0 for j = i and j = i 1,
¯i j − j ¯i 6 6 −
q2ki q 2ki
(2.3) k¯i2 = q2−q−2 , k¯ik¯j = −k¯jk¯i for i6= j,
−
−
e¯if¯i+f¯ie¯i = qki+ki+q1 −qq−1ki−ki+1 +(q−q−1)k¯iki+1,
−
−
qe f +f e = e f +qf e = e f +f e = 0 if i j > 1,
i+1 ¯i ¯i i+1 ¯i i+1 i+1 ¯i ¯i ¯j ¯j ¯i | − |
k¯ie¯i+qe¯ik¯i = eiq−ki, qk¯iei 1+ei 1k¯i = q−kiei 1,
− − −
k e +e k = 0 for j = i and j = i 1,
¯i ¯j ¯j ¯i 6 6 −
k¯if¯i+qf¯ik¯i = fiqki, qk¯ifi 1+fi 1k¯i = qkifi 1,
− − −
k f +f k = 0 for j = i and j = i 1,
¯i ¯j ¯j ¯i 6 6 −
q q 1 q q 1
e2 = − − e2, f2 = − − f2,
¯i −q+q 1 i ¯i q+q 1 i
− −
e e e e = f f f f = e e +e e = f f +f f = 0 if i j > 1,
i j − j i i j − j i ¯i ¯j ¯j ¯i ¯i ¯j ¯j ¯i | − |
e e e e = f f f f = 0 if i j = 1,
i ¯j − ¯j i i ¯j − ¯j i | − | 6
e e e e = e e +e e , f f f f = f f +f f ,
i i+1− i+1 i ¯i i+1 i+1 ¯i i+1 i− i i+1 ¯i i+1 i+1 ¯i
e e e e = e e e e , f f f f = f f f f ,
i i+1− i+1 i ¯i i+1− i+1 ¯i i+1 i− i i+1 i+1 ¯i− ¯i i+1
qe2e (q+q 1)e e e +q 1e e2 = 0,
i i+1− − i i+1 i − i+1 i
qf2f (q+q 1)f f f +q 1f f2 = 0,
i i+1− − i i+1 i − i+1 i
10 GRANTCHAROV,JUNG,KANG,KIM
qe e2 (q+q 1)e e e +q 1e2 e = 0,
i i+1− − i+1 i i+1 − i+1 i
qf f2 (q+q 1)f f f +q 1f2 f = 0,
i i+1− − i+1 i i+1 − i+1 i
qe2e (q+q 1)e e e +q 1e e2 = 0,
i i+1− − i i+1 i − i+1 i
qf2f (q+q 1)f f f +q 1f f2 = 0,
i i+1− − i i+1 i − i+1 i
qe e2 (q+q 1)e e e +q 1e2 e = 0,
i i+1− − i+1 i i+1 − i+1 i
qf f2 (q+q 1)f f f +q 1f2 f = 0.
i i+1− − i+1 i i+1 − i+1 i
Proof. LetU betheunitalassociative algebraover C(q)generated bytheelements ei,fi,e¯i,f¯i
(i = 1,...,n 1), k (l = 1,...,n), andqh (h P )withdefiningrelations given in(2.3). Using
− ¯l ∈ ∨
(2.1)and(2.2),therelationsin(2.3)canbederivedeasily. Thusthereisawell-definedalgebra
homomorphism φ :U U (g).
q
−→
From the relation (2.1), we obtain
j 1
Li,i+j = (q q−1)q−Pjh−=11ki+h − adfi+h(fi),
−
h=1
Y
j 1
L−i,i+j = −(q−q−1)q−Pjh−=11ki+h − adfi+h(f¯i),
h=1
(2.4) Y
j 1
L−i−j, i = (−1)j(q−q−1)qPjh−=11ki+h − adei+h(e¯i),
h=1
Y
j 1
L i j, i = ( 1)j(q q−1)qPjh−=11ki+h − adei+h(ei),
−− − − −
h=1
Y
whereadb (b ) := b b b b , j adb (b ) := adb adb (b )and 0 adb (b ) =
i j i j− j i h=1 i+h i i+j··· i+1 i h=1 i+h i
b for b = e ,e ,f ,f (i = 1, ,n 1, j > 0). It follows that the homomorphism φ must
i i i ¯i i ¯i Q··· − Q
be surjective.
It remains to prove φ is injective. For this purpose, we will show that the relations in (2.1)
can bederived from the ones in (2.3). Theproofof our assertion is quite lengthy and tedious.
But the basic idea is just the case-by-case check-up.
We define the sets
Λ= (i,j) Z/ 0 Z/ 0 n i j n , Λ = (i,j) Λ i > 0,j > 0 and i< j ,
1
{ ∈ { }× { } | − ≤ ≤ ≤ } { ∈ | }
Λ = (i,j) Λ i < 0,j > 0 and i < j , Λ = (i,j) Λ i < 0,j > 0 and i > j ,
2 3
{ ∈ | | | | |} { ∈ | | | | |}
Λ = (i,j) Λ i < 0,j < 0 and i > j , Λ = (i,j) Λ i < 0,j > 0 and i = j .
4 5
{ ∈ | | | | |} { ∈ | | | | |}