Two-parameter quantum general linear supergroup and central extensions

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by  Huafeng Zhang

TWO-PARAMETER QUANTUM GENERAL LINEAR SUPERGROUP
AND CENTRAL EXTENSIONS
HUAFENGZHANG
6
1
0 Abstract. TheuniversalR-matrixoftwo-parameterquantumgenerallinearsupergroups
2 is computed explicitly based on the RTT realization of Faddeev–Reshetikhin–Takhtajan.

n For the two-parameter quantum special linear supergroup associated with sl(2,2), a two-
a fold quasi-central extension is deduced from its Drinfeld–Jimbo realization.
J

9
2

1. Introduction
]
A
Let r,s be non-zero complex numbers whose ratio r is not a root of unity. Let M,N be
Q s
positive integers and g := gl(M,N) be the general linear Lie superalgebra associated to the
h. vector superspace CM N. The universal enveloping algebra of U(g) as a Hopf superalgebra
|
t admits a two-parameter deformation U (g) which is neither commutative nor cocommuta-
a r,s
m tive. In this paper we compute its universal R-matrix, an invertible element in a completed
[ tensor square R ∈Ur,s(g)⊗b2 satisfying among other favorable properties (Theorem 4.1)

1 ∆cop(x) = R∆(x)R 1 for x ∈ U (g).
− r,s
v
6 There are explicit formulas of universal R-matrices for one-parameter quantum groups
9 associated to simple Lie (super)algebras, quantum affine algebras and double Yangians,
1
and certain quantum Kac–Moody superalgebras (without isotropic roots), to name a few;
8
0 usually they are realized as the tensor product of orthonormal bases with respect to a
. bilinear form called the Drinfeld quantum double (or Hopf pairing in the main text). They
1
0 have important applications in quantum integrable systems, knot invariants, etc.
6 For the two-parameter quantum group U (gl(M)), Benkart–Witherspoon [BW1, BW2]
r,s
1
proved the existence of universal R-matrix, which leads to a braided structure of a category
:
v O of representations; the exact formula of universal R-matrix was unknown. There are
Xi two equivalent definitions of U (gl(M)): the Drinfeld–Jimbo realization, and the RTT
r,s
r realization of Faddeev–Reshetikhin–Takhtajan [FRT, JL].
a
In [Z2] for the quantum affine superalgebra associated to gl(1,1) the universal R-matrix
R was computed. The main steps therein were: first relate the quantum double to the RTT
realization; then obtain orthonormal bases (and R) via a Gauss decomposition in [DF].
Motivated by [BW1, JL, Z2], in the present paper we start with the RTT realization of
U (g), based on a suitable solution of the quantum Yang–Baxter equation on CM N.
r,s |
Our first main result is a factorization formula of R in Equations (4.6)–(4.8). This
is related to the fact that the orthonormal bases are formed of ordered products. We
would like to emphasize that: within the framework of the RTT realization of U (g),
r,s
the orthogonality of these ordered products is fairly easy to prove, compared to the usual
Drinfeld–Jimbo realization in [BW1]; see Remark 3.5. Making use of the quantum double
1
2 HUAFENGZHANG
we are able to simplify some computations of relations between RTT generators which were
done in [JL] almost case by case; see the proof of Corollary 2.8.
Our second main result is related to the special linear Lie superalgebra sl(2,2), which
fills in a non-split short exact sequence of Lie superalgebras C2 ֌ h ։ sl(2,2) with C2
being central in h [IK]. TheLie superalgebra h is a two-fold central extension of sl(2,2). Its
finite-dimensional representation theory plays a special roˆle in the integrability structure of
the anti-de Sitter/conformal field theory correspondence; see for example [Be, MM]. Not
surprisingly, the quantum supergroupU (sl(2,2)) admits a two-fold central extension U (h)
q q
[Y2], by forgetting the degree 4 oscillator relations in [Y1].
We propose in Definition 6.1 a two-parameter quantum supergroup U associated to
r,s
h. Namely we construct a surjective morphism of Hopf superalgebras U ։ U (sl(2,2))
r,s r,s
whose kernel is generated by two elements P,Q which are “central up to powers of rs”; see
Proposition 6.3. The quantum supergroup U (h) in [Y2] is a quotient of U .
q q,q−1
Whether rs is a root of unity or not has an impact on the representation theory of U .
r,s
We construct (Proposition 6.5 and Corollary 6.6) a one-parameter family of irreducible
representations (π ) of the U by modifying the vector representation of U (sl(2,2)) on
x r,s r,s
C22; the additional parameter x ∈ C arises from the dynamical action of P.
| ×
This paper is organized as follows. §2 provides the RTT realization of U (g). In §3 an
r,s
orthogonal property of the quantum double is proved. The universal R-matrix R is written
down in §4 and a category O of representations is introduced in order to specialize R.
§5 presents another system of generators for U (g), the Drinfeld–Jimbo realization, from
r,s
which we deduce easily a two-fold quasi-central extension U in §6, whose representation
theory is initiated. The final section §7 is left to further discussions.
2. Yang–Baxter equation and RTT realization
In this section we provide a solution of the quantum Yang–Baxter equation and use it to
define U (gl(M,N)) following Faddeev–Reshetikhin–Takhtajan [FRT].
r,s
LetV = CM N bethevector superspacewithbasis(v ) andparity: |v | = |i| = 0
| i 1 i M+N i
if i ≤ M and |v | = |i| = 1 if i > M. Since r+s 6= 0, the≤te≤nsor square V 2 is easily seen
i ⊗
to be a direct sum of the following two sub-vector-superspaces:
S2V := Vect(v 2,v ⊗v +(−1)j ksv ⊗v | i ≤ M,j < k),
i⊗ j k | || | k j
∧2V := Vect(v 2,v ⊗v −(−1)j krv ⊗v | i > M,j < k).
i⊗ j k | || | k j
LetP,Qbethecorrespondingprojections. Definethetwo-parameter Perk–Schultz matrices
(2.1) R := rP −sQ∈ End(V⊗2), R := cV,VR ∈ End(V⊗2)
where cV,V ∈ End(V⊗2) : vi ⊗ vj 7→ (−1)|i||j|vj ⊗ vi is the graded permutation. Let
b b
E ∈ EndV be v 7→ δ v . The precise formula of R is
ij k jk i
(2.2) R = (r +s )E ⊗E +( +rs )E ⊗E +(r−s) (−1)iE ⊗E .
ii ii ii jj || ji ij
i M i>M i>j i<j i<j
X≤ X X X X
Let A,B,C be superalgebras and let t = a ⊗ b ∈ A ⊗ B. Define the tensors t =
i i i 12
t⊗ 1, t = 1⊗t, t = a ⊗1⊗b with 1 ∈ C the identity element. The following
23 13 i i i
P
quantum Yang–Baxter equation is a consequence of the non-graded case N = 0 in [BW2].
P
QUANTUM SUPERGROUP 3
Lemma 2.1. R R R = R R R ∈ (EndV) 3.
12 13 23 23 13 12 ⊗
Proof. By Equation (2.1) this is equivalent to the braid relation for R:
(∗) R R R = R R R ∈ (EndV) 3.
12 23 12 23 12 23 ⊗ b
Claim 1. (∗) is true if N = 0 or M = 0.
b b b b b b
When N = 0, the proof of [BW2, Proposition 5.5] indicates that P − rs 1Q satisfies
−
(∗); so does R = r(P −rs 1Q) 1. The case of M = 0 comes from that of N = 0 with
− −
parameters (s,r) instead of (r,s) owing to Equation (2.2); see also Remark 2.3.
Claim 2. (∗) ibs true when applied to the vector va⊗vb⊗vc with ♯{a,b,c} = 3.
Let S be the matrix R associated to CM+N 0, so that S satisfies (∗). Let Sij be the
| i′j′
coefficient of vi′ ⊗vj′ in S(vi ⊗vj). Then Siij′j′ 6= 0 only if {i,j} = {i′,j′}. When i 6= j,
b
R(v ⊗v ) = Sijv ⊗v +(−1)i j Sijv ⊗v .
i j ij i j ||| | ji j i
Now assume 1 ≤ a,b,c ≤ M +N two-by-two distinct. Write
b
S12S23S12(va⊗vb⊗vc)= Caa′bbc′c′va′ ⊗vb′ ⊗vc′ = S23S12S23(va⊗vb⊗vc).
a′,b′,c′
X
Then Cabc 6= 0 only if {a,b,c} = {a,b,c}. Furthermore, one can find εabc = ± depend-
a′b′c′ ′ ′ ′ a′b′c′
ing only on the permutation (a,b,c) of (a,b,c) such that
′ ′ ′
R12R23R12(va⊗vb⊗vc)= εaa′bbc′c′Caa′bbc′c′va′ ⊗vb′ ⊗vc′ = R23R12R23(va⊗vb⊗vc).
a′,b′,c′
X
b b b b b b
The same arguments can be used to verify the case where: (a,b,c) is a permutation of
(i,i,j) with i ≤ M. We are led to three remaining cases.
Claim 3. For M = N = 1, (∗) applied to v ⊗v ⊗v ,v ⊗v ⊗v ,v ⊗v ⊗v is true.
1 2 2 2 1 2 2 2 1
Consider the middle vector v ⊗v ⊗v =:u for example:
2 1 2
R R R (u) = R R (v ⊗v ⊗v )= −sR (v ⊗v ⊗v )
12 23 12 12 23 1 2 2 12 1 2 2
= −s(r−s)v ⊗v ⊗v −rs2v ⊗v ⊗v
1 2 2 2 1 2
b b b b b b
= R ((r−s)v ⊗v ⊗v −rs2v ⊗v ⊗v )
23 1 2 2 2 2 1
= R R ((r−s)v ⊗v ⊗v +rsv ⊗v ⊗v ) = R R R (u).
23 12 2 1 2 2 2 1 23 12 23
b
The other two vectors can be checked in the same way. (cid:3)
b b b b b
We are ready to associate a two-parameter quantum supergroup to gl(M,N).
Definition 2.2. U := U (gl(M,N)) is the superalgebra generated by the coefficients of
r,s
matrices T = t ⊗E ,S = s ⊗E ∈ U⊗EndV of even parity with relations
i j ji ji i j ij ij
≤ ≤
R TPT = T T R , PR S S = S S R , R T S = S T R ,
23 12 13 13 12 23 23 12 13 13 12 23 23 12 13 13 12 23
and with requirement that the s ,t be invertible for 1≤ i ≤ M +N.
ii ii
4 HUAFENGZHANG
Remark 2.3. Let R′ := cV,VR−1cV,V. The above three equations are equivalent to
R T T = T T R , R S S = S S R , R S T = T S R .
2′3 12 13 13 12 2′3 2′3 12 13 13 12 2′3 2′3 12 13 13 12 2′3
Let us write RrM,s|N,Ur,s when the dependence on r,s,M,N is relevant. rsRr′n,s|0 is exactly the
R-matrix used in [JL, Definition 3.1] to define the two-parameter quantum group U (gl ).
r,s n
0n n0
Since Rr,|s = Rs,|r, there are algebra morphisms from Ur,s(glM) and Us,r(glN) in [JL] to our
U , the former being l+ 7→ s ,l 7→ t and the latter l+ 7→ s ,l 7→ t .
r,s ij ij j−i ji ij M+i,M+j j−i M+j,M+i
(We borrowed the notations l+,l from that paper.)
ij j−i
U is a Hopf superalgebra with coproduct ∆ and counit ε:
∆(s )= s ⊗s , ∆(t ) = t ⊗t , ε(s ) = ε(t ) =δ .
ij ik kj ji jk ki ij ji ij
k k
X X
The antipode S : U −→ U is an anti-automorphism of superalgebra defined by equations
(S⊗Id)(S) = S 1,(S⊗Id)(T) = T 1 in U⊗EndV. Let U+ (resp. U ) bethe subalgebra of
− − −
U generated by the s ,s 1 (resp. the t ,t 1) for i≤ j; these are sub-Hopf-superalgebras.
ij −kk ji −kk
Proposition 2.4. There exists a unique Hopf pairing ϕ :U+×U −→ C such that
−
ϕ(s ,t )E ⊗E = R ∈ (EndV) 2.
ij kl kl ij ⊗
ijkl
X
The associated quantum double D = U+ ⊗U is isomorphic to U as Hopf superalgebras
ϕ −
via the multiplication map U+⊗U −→ U,x⊗y 7→ xy. (cid:3)
−
Remark 2.5. ϕ being a Hopf pairing means that: for a,a ,a ∈U+ and b,b ,b ∈U
1 2 1 2 −
ϕ(a,b b )= ϕ (∆(a),b ⊗b ), ϕ(a a ,b) = (−1)a1 a2 ϕ (a ⊗a ,∆(b)),
1 2 2 1 2 1 2 | || | 2 2 1
here ϕ (a ⊗a ,b ⊗b ) = (−1)a2 b1 ϕ(a ,b )ϕ(a ,b ). The quantum double means
2 1 2 1 2 | || | 1 1 2 2
ba= (−1)|a(1)||b|+(|b(2)|+|b(3)|)|a(2)|+|a(3)||b(3)|ϕ(a(1),S(b(1)))a(2)b(2)ϕ(a(3),b(3)).
We refer to [Z1] for a sketch of proof of the above proposition.
Proposition 2.6. The assignment s 7→ ε t , t 7→ ε s extends uniquely to an iso-
ij ji ji ji ij ij
morphism of Hopf superalgebras U −→ Ucop . Here ε := (−1)i+i j .
r,s s−1,r−1 ij || ||| |
Proof. Let τ : EndV −→ EndV be the super transposition E −→ ε E . Then
ij ij ji
τ⊗2(Rr−,s1)= r−1s−1cV,VRs−−11,r−1cV,V.
Therestisclear fromDefinition 2.2, asintheone-parameter case[Z1]. Heretheco-opposite
Acop of a Hopf superalgebra (A,∆,ε,S) is by definition (A,c ∆,ε,S 1). (cid:3)
A,A −
Lemma 2.7. There is a vector representation ρ of U on V defined by
(ρ⊗1)(S) =(τ ⊗1)(R), (ρ⊗1)(T) = rs(τ ⊗1)(cV,VR−1cV,V).
Proof. Straightforward using Lemma 2.1 and Definition 2.2. (cid:3)
QUANTUM SUPERGROUP 5
Corollary 2.8. Let 1≤ i,j,k ≤ M +N be such that j ≤ k. Then
s s = ϕ(s ,t )ϕ(s ,t ) 1s s , t s = ϕ(s ,t ) 1ϕ(s ,t )s t ,
ii jk ii jj ii kk − jk ii ii jk jj ii − kk ii jk ii
t t = ϕ(s ,t )ϕ(s ,t ) 1t t , s t = ϕ(s ,t ) 1ϕ(s ,t )t s .
ii kj jj ii kk ii − kj ii ii kj ii jj − ii kk kj ii
Proof. For the second equation, by Proposition 2.4 and Remark 2.5
t s = ϕ(s ,S(t ))s t ϕ(s ,t )= ϕ(s ,t ) 1ϕ(s ,t )s t .
ii jk jj ii jk ii kk ii jj ii − kk ii jk ii
Here wehave usedthethree-fold coproductformula of t ,s , and thefact that ϕ(s ,t )=
ii jk ab ii
0 if a < b. The fourth equation can be proved in the same way.
For the first equation, by comparing the coefficients of v ⊗v in the identical vectors
i j
R S S (v ⊗v ) = S S R (v ⊗v )∈ U⊗V 2 we obtain (let s = 0 if p > q)
23 12 13 i k 13 12 23 i k ⊗ pq
xs s +ys s = zs s +ws s
ii jk ji ik jk ii ji ik
for certain x,z ∈ {1,rs,r,s} and y,w ∈ {0,r −s,s−r}. We prove that s s ∈ Cs s .
ii jk jk ii
If not, then j < i < k, in which case y = (−1)i (r − s) = w and xs s = zs s , a
|| ii jk jk ii
contradiction. Now the first equation is obtained from the representation ρ in Lemma 2.7:
ρ(s )= ϕ(s ,t )E , ρ(s )= ϕ(s ,t )ε E for j < k.
ii ii kk kk jk jk kj kj jk
k
X
The third equation can be proved in the same way. (cid:3)
Let P:= ⊕M+NZǫ be the weight lattice. Call a vector x ∈ U of weight λ ǫ ∈P if
i=1 i i i i
(2.3) siixs−ii1 = ϕ(sii,tii)λi(rs)Pj<iλjx, tiixt−ii1 = ϕ(sii,tii)−λi(rs)Pj≤Piλjx.
s and t are of weight ±(ǫ −ǫ ) respectively. The Hopf superalgebra U is P-graded.
ij ji i j
3. Orthogonal property
In this section we prove an orthogonal property of the Hopf pairing ϕ : U+ ×U −→ C
−
in Proposition 2.4; it is eventually a consequence of RTT in Definition 2.2.
Let us introduce the following four subalgebras of U :
±
U+0 := hs 1 | 1 ≤ i≤ M +Ni, U> := ha := s 1s | 1 ≤ i< j ≤ M +Ni ⊆ U+,
±ii ij −ii ij
U 0 := ht 1 | 1 ≤ i≤ M +Ni, U< := hb := t t 1 | 1 ≤ i < j ≤M +Ni⊆ U .
− ±ii ji ji −ii −
U+ = U+0U> and U = U 0U< by Corollary 2.8.
− −
Lemma 3.1. ϕ(x a,x b) = ϕ(x ,x )ϕ(a,b) for a ∈ U>,b ∈ U< and x ∈U 0.
+ + ±
− − ±
Proof. Observethat: if x ∈ U+ andy ∈ U areof weights α,β respectively, then ϕ(a,b) 6= 0
−
only if α + β = 0. One may assume that x is a product of the s 1. By definition,
+ ±ii
∆(x a)−x ⊗x a is a sum of x ⊗y where the weights of the x are Z -spans of the
+ + + i i i >0
ǫ −ǫ with k < l; which forces ϕ(x ,x ) = 0. By Remark 2.5,
k l i
−
ϕ(x a,x b) = ϕ (∆(x a),x ⊗b)= ϕ (x ⊗x a,x ⊗b) = ϕ(x ,x )ϕ(x a,b).
+ 2 + 2 + + + +
− − − −
∆(b)−b⊗1 is a sum of x ⊗y with y being of non-zero weight and ϕ(x ,y ) = 0. So
′i i′ i′ + i′
ϕ(x a,b) = ϕ (a⊗x ,∆(b)) =ϕ (a⊗x ,b⊗1) = ϕ(a,b)ϕ(x ,1) = ϕ(a,b).
+ 2 + 2 + +
Here ϕ(x ,1) = 1 by our assumption on x . This gives the desired formula. (cid:3)
+ +
6 HUAFENGZHANG
Endow the (1 ≤ i< j ≤ M +N) with the order: (i,j) ≺ (k,l) if (i < k) or (i = k,j <l),
in which case write a ≺ a and b ≺ b . This defines a total order on
ij kl ji lk
X := {a | 1 ≤ i< j ≤ M +N} ⊆ U>, Y := {b | 1≤ i < j ≤ M +N} ⊆U<.
ij ji
Lemma 3.2. Fix 1 ≤ i < j ≤ M + N and p ∈ Z . Let x ,x ,··· ,x ∈ X and
>0 1 2 p
y ,y ,··· ,y ∈ Y be such that x (cid:23) a and y (cid:23) b for all 1≤ l ≤ p.
1 2 p l ij l ji
(1) ϕ(a ,b ) = (−1)i(s 1−r 1).
ij ji || − −
(2) If ϕ(a ,y y ···y ) 6= 0, then p = 1,y = b .
ij 1 2 p 1 ji
(3) If ϕ(x x ···x ,b )6= 0, then p = 1 and x = a .
1 2 p ji 1 ij
Proof. We compute ϕ(a ,b ) by using Proposition 2.4, Corollary 2.8 and Lemma 3.1:
ij ji
(−1)i(r−s) = ϕ(s ,s ) = ϕ(s a ,b t )= ϕ(s a ,ϕ(s ,t ) 1ϕ(s ,t )t b )
|| ij ji ii ij ji ii ii ij ii ii − jj ii ii ji
= ϕ(s ,t )ϕ(a ,b ) = rsϕ(a ,b ). =⇒ (1).
jj ii ij ji ij ji
For (2), notice that the first tensor factors in ∆(a )−a ⊗s 1s , being either 1 or x ∈ X
ij ij −ii jj
withx ≺a ,areorthogonaltoy (cid:23) b . Soϕ(a ,y y ···y ) = ϕ(a ,y )ϕ(s 1s ,y ···y ).
ij 1 ji ij 1 2 p ij 1 −ii jj 2 p
In view of the weight grading, p = 1 and y = b . (3) is proved in the same way. (cid:3)
1 ji
Corollary 3.3. Fix 1 ≤ i < j ≤ M +N. Let x ,x ,··· ,x ∈ X and y ,y ,··· ,y ∈ Y be
1 2 p 1 2 q
such that x ≻ a and y ≻ b for all l. Let m,n ∈ Z . Then
l ij l ji 0
≥
ϕ(x x ···x am,y y ···y bn)= ϕ (x x ···x ⊗am,y y ···y ⊗bn),
1 2 p ij 1 2 q ji 2 1 2 p ij 1 2 q ji
ϕ(am,bn)= δ (m)! ϕ(a ,b )m.
ij ji mn τij ij ji
Here (m) = um 1,(m)! = m (k) and τ = (−1)i+j (rs) 1ϕ(s ,t )ϕ(s ,t ).
u u −1 u k=1 u ij || | | − ii ii jj jj
−
Proof. Use induction on maxQ(m,n): the case m = n= 0 is trivial. Assume m > 0 (the case
n > 0 can be treated similarly). The left hand side of the first formula becomes
lhs = (−1)θ1ϕ (a ⊗x x ···x am 1,∆(y y ···y bn)), θ = |a ||am 1x x ···x |.
1 2 ij 1 2 p ij− 1 2 q ji 1 ij ij− 1 2 p
Notice that the first tensor factors of ∆(b )−b ⊗1−t t 1⊗b are of the form xy where
ji ji jj −ii ji
x ∈ U 0,y ∈ Y such that y ≻ b and ϕ(1,x) = 1. By Lemmas 3.1–3.2,
− ji
q n
lhs1 = (−1)θ1ϕ2(aij ⊗x1x2···xpamij−1, (zk ⊗yk) (z⊗bji)l−1(bji ⊗1)(z ⊗bji)n−l).
k=1 l=1
Y X
Here z = t t 1,z ∈ U 0 and ϕ(1,z )= 1 for 1 ≤ k ≤ q. By Corollary 2.8,
jj −ii k − k
b z = zb ϕ(s ,t ) 1ϕ(s ,t ) 1ϕ(s ,t )ϕ(s ,t )= zb τ (−1)bji .
ji ji ii jj − jj ii − ii ii jj jj ji ij | |
Thus (z⊗bji)l−1(bji⊗1)(z ⊗bji)n−l = (−1)(n−1)|bji|τinj−lzn−1bji⊗bjni−1 and
lhs = (−1)θ1+θ2(n) ϕ (a ⊗x x ···x am 1,z z ···z zn 1b ⊗y y ···y bn 1)
1 τij 2 ij 1 2 p ij− 1 2 q − ji 1 2 q ji−
= (−1)θ1+θ2(n) ϕ (a ⊗x x ···x am 1,b ⊗y y ···y bn 1)
τij 2 ij 1 2 p ij− ji 1 2 q ji−
= (n) ϕ (x x ···x am 1⊗a ,y y ···y bn 1⊗b ).
τij 2 1 2 p ij− ij 1 2 q ji− ji
Here θ = |b ||bn 1y y ···y |. In the last identity ϕ (a ⊗b,c ⊗ d) = ϕ (b ⊗ a,d ⊗ c)×
2 ji ji− 1 2 q 2 2
(−1)a b+c d as ϕ respects the parity. The rest is clear from the induction hypothesis. (cid:3)
| || | | || |
QUANTUM SUPERGROUP 7
Let Γ be the set of functions f : X −→ Z such that f(x) ≤ 1 if |x| = 1. Such an f
0
induces, by abuse of language, another functio≥n f : Y −→ Z with f(b ) := f(a ). Set
0 ji ij
≥
≻ ≻
(3.4) a[f]:= xf(x) ∈ U>, b[f]:= yf(y) ∈ U<.
x X y Y
Y∈ Y∈
Here ≻ means the product with descending order. If i ≤ M < j, then τ = −1 and
ij
ϕ(am,bm) = 0 for m > 1, which is the reason for f(a ) ≤ 1.
ij ji ij
Q
Corollary 3.4. For f,g ∈ Γ, ϕ(a[f],b[g]) 6= 0 if and only if f = g. Moreover, the a[f] and
the b[f] form a basis of U> and U< respectively.
Proof. Thefirststatement comes fromCorollary 3.3andLemma3.2; notably thea[f](resp.
the b[f]) are linearly independent. For the second statement, consider U> for example. A
slightmodificationoftheargumentsintheproofof[MRS,Lemma2.1]byusingR S S =
23 12 13
S S R shows that U> is spanned by odered monomials of the a . It suffices to check
13 12 23 ij
that s2 = 0 (and so a2 =0) for i ≤ M < j; this comes from a comparison of coefficients of
ij ij
v ⊗v in the identical vectors R S S (v ⊗v )= S S R (v ⊗v )∈ U⊗V 2. (cid:3)
i i 23 12 13 j j 13 12 23 j j ⊗
Remark 3.5. For the one-parameter quantum supergroup U (gl(M,N)), there is a Hopf
q
pairing with an orthogonal property [Y1, §10.2] similar to Corollary 3.4. The proof in loc.
cit. was a lengthy calculation on coproduct estimation and commuting relations of root
vectors defined as proper quantum brackets of Drinfeld–Jimbo generators; see also [R, §II].
In our situation, the RTT generators can be viewed as root vectors; their coproduct and
commuting relations are given for free in the definition.
4. Universal R-matrix
Inthis section we computetheuniversal R-matrix of U . For this purpose,wefirstwork
r,s
with a topological version of quantum supergroups and view r,s as formal variables:
r = e~ ∈ C[[~,℘]], s = e℘ ∈ C[[~,℘]].
Step 1. Extend U ,U to topological Hopf superalgebras over C[[~,℘]] according to Equa-
±
tion (2.3) : first add commutative primitive elements (ǫ ) of even parity such that
∗i 1 i M+N
[ǫ ,x]= λ x for x ∈ U ,U of weight λ = λ ǫ ∈ P; then≤i≤dentify
∗i i ± i i i
(4.5) sii = e(~+℘)Pj<iǫ∗j ×(ee℘~ǫǫ∗i∗i ((ii >≤PMM)),, tii = e(~+℘)Pj<iǫ∗j ×(ee~℘ǫǫ∗i∗i ((ii >≤ MM)).,
Denote by U±~,℘,U~,℘ the resulting topological Hopf superalgebras. They contain
℘ (i ≤ M),
H := (~+℘) ǫ +ǫ
i ∗j ∗i (~ (i > M).
j<i
X
Extend ϕ to a Hopf pairing ϕ : U+ ×U −→ C((~,℘)) by ϕ(ǫ ,H ) = δ . Observe that
~,℘ −~,℘ ∗i j ij
ϕ(s ,t ) = ϕ(s ,t ), which shows in turn that ϕ exists uniquely. As in Proposition 2.4,
ii jj ii jj
the multiplication map induces a surjective morphism of topological Hopf superalgebras
from the quantum double Dϕ to U~,℘ with kernel generated by the ǫ∗i ⊗1−1⊗ǫ∗i.
8 HUAFENGZHANG
Step 2. Let U0 be the topological subalgebra of U generated by the ǫ . Then U+ =
±~,℘ ∗i ~,℘
U0U> and U = U0U<. Lemma 3.1 holds true for x ∈ U0. Together with Corollaries
−~,℘
3.3–3.4 we obtain orthonormal bases of ϕ and the unive±rsal R-matrix of U~,℘:
a[f]⊗b[f] ≻
(4.6) R = R R , R := (−1)a[f] = R ,
0 + + | | ij
ϕ(a[f],b[f])
f Γ 1 i<j M+N
X∈ ≤ Y≤
M+N M M+N
(4.7) R0 := eǫ∗i⊗Hi = sǫ∗i⊗ǫ∗i × rǫ∗j⊗ǫ∗j × (rs)ǫ∗k⊗ǫ∗l ,
   
!
i=1 i=1 j=M+1 1 l<k M+N
Y Y Y ≤ Y≤
∞ anij⊗bnji if (i <j ≤ M),   
(n)! (s−1 r−1)n
n=0 rs−1 −

(4.8) Rij = nP∞=0(n)!sr−a1ni(jr⊗−b1nj−is−1)n if (M < i< j),
1P− aij⊗bji if (i ≤ M < j).
s−1 r−1
 −
The formula of R is similar to that for Uq(gl(M,N)) in [Y1, §10.6] when r = q = s−1. We
shall evaluate R in a certain class of representations (defined over C). From now on assume
that r,s ∈ C× and rs is not a root of unity. We work with Ur,s = U instead of U~,℘.
Step 3. Let V be a U-module and 0 6= v ∈ V. Call v a weight vector of weight λ ∈ P if
(4.9) siiv = ϕ(sii,tii)λi(rs)Pj<iλjv, tiiv = ϕ(sii,tii)−λi(rs)Pj≤iλjv.
Call V a weight module if it is spanned by weight vectors. Define Q (resp. Q+) to be the
Z-span (resp. the Z -span) of the ǫ −ǫ for 1 ≤ i < j ≤ M +N. As in [BW1], V is said
0 i j
to be in category O≥if it is a weight module with finite-dimensional weight spaces and
(BGG) the weights of V are contained in ∪ (λ−Q+) for some finite subset F ⊂ P.
λ F
Let V,W be in category O. Firstly (R ) ∈∈End(V ⊗W) is well-defined by
0 V,W
v⊗w 7→ v⊗w×sPi≤MλiµirPj>Mλiµi(rs)Pk>lλkµl
for v ∈ V and w ∈ W being of weight λ = λ ǫ and µ = µ ǫ respectively. (In the
i i i i i i
case (M,N) = (n,0), this is exactly the operator s × f in [BW1, §4].) Secondly, let
P V,W P
v ∈ V is of weight λ ∈ P. For f ∈ Γ, by Equations (2.3) and (3.4), the vector a[f]v ∈ V is
of weight λ+ i<jf(aij)(ǫi −ǫj). By (BGG), a[f]v = 0efor all but finite f. This implies
that (R ) ∈ End(V ⊗W) is also well-defined. Let R := (R ) (R ) . From the
+ V,W V,W 0 V,W + V,W
quantum doubPle construction of U, we obtain (compare [BW1, Theorems 4.11, 5.4])
Theorem 4.1. Let V ,V ,V be in category O. Then
1 2 3
(R ) (R ) (R ) = (R ) (R ) (R ) ∈ End(V ⊗V ⊗V )
V1,V2 12 V1,V3 13 V2,V3 23 V2,V3 23 V1,V3 13 V1,V2 12 1 2 3
and c R : V ⊗V −→ V ⊗V is an isomorphism of U-modules. (cid:3)
V1,V2 V1,V2 1 2 2 1
Example 4.2. Consider the vector representation of U on V in Lemma 2.7. From the
proof of Corollary 2.8 we see that v is of weight ǫ and V is in category O. Moreover,
i i
ρ(s )= E (r−s)(−1)k , ρ(t )= E (s−r)(−1)k (j < k).
jk jk | | kj kj | |
This gives RV,V = cV,VRs−−11,r−1cV,V. Compare [Y1, §10.7].
QUANTUM SUPERGROUP 9
5. Drinfeld–Jimbo realization
We present the Drinfeld–Jimbo realization of U (gl(M,N)). As in [JL, §4], let us intro-
r,s
duce Drinfeld–Jimbo generators: for 1 ≤ i< M +N
E := a = s 1s , F := b = t t 1, K := s 1s , L := t t 1
i i,i+1 −ii i,i+1 i i+1,i i+1,i −ii i −ii i+1,i+1 i i+1,i+1 −ii
It follows that K ,L are grouplike and E ,F are skew-primitive:
i i i i
(5.10) ∆(E ) = 1⊗E +E ⊗K , ∆(F )= L ⊗F +F ⊗1, ∆(g) = g⊗g for g = K ,L .
i i i i j j j j i i
Lemma 5.1. s s −s s = (−1)b(r−s)s s for 1 ≤ a < b < c ≤ M +N.
ab bc bc ab | | ac bb
Proof. Straightforward by comparing the coefficients of v ⊗v in the two identical vectors
b a
R S S (v ⊗v ) = S S R (v ⊗v ) ∈ U⊗V 2. (cid:3)
23 12 13 c b 13 12 23 c b ⊗
Proposition 5.2. U is generated as a superalgebra by the E ,F ,s 1,t 1. Moreover:
i i ±jj ±jj
(5.11) E2E −(r+s)E E E +rsE E2 = 0 if (1 ≤ i < M +N −1,i 6= M),
i i+1 i i+1 i i+1 i
(5.12) E E2−(r+s)E E E +rsE2E = 0 if (1 < i < M +N,i 6= M),
i 1 i i i 1 i i i 1
− − −
(5.13) rsF2F −(r+s)F F F +F F2 = 0 if (1 ≤ i < M +N −1,i 6= M),
i i+1 i i+1 i i+1 i
(5.14) rsF F2−(r+s)F F F +F2F = 0 if (1 < i < M +N,i 6= M),
i 1 i i i 1 i i i 1
− − −
(5.15) E E = E E , F F = F F , E2 = F2 = 0 if (|i−j| > 1),
i j j i i j j i M M
(5.16) [E ,F ] = δ (−1)i(s 1−r 1)(K −L ).
i j ij || − − i i
When M,N > 1 the following relations also hold true:
E E E E +rsE E E E +E E E E
(5.17) M−1 M M+1 M M+1 M M−1 M M M−1 M M+1
+ rsE E E E −(r+s)E E E E = 0.
M M+1 M M 1 M M 1 M+2 M
− −
rsF F F F +F F F F +rsF F F F
(5.18) M−1 M M+1 M M+1 M M−1 M M M−1 M M+1
+ F F F F −(r+s)F F F F = 0.
M M+1 M M 1 M M 1 M+2 M
− −
Proof. (Sketch) This is based on lengthy but straightforward calculations as done in the
non-graded case in [JL, §4]. Equations (5.11)–(5.16) agree with [JL, Equations (4.17),
(4.19)–(4.21), (4.24)–(4.26)] by Remark 2.3. We only consider Equations (5.16) and (5.17).
From Remark 2.5, Lemma 3.1 and Equation (5.10) we obtain
F E = (−1)Ei Fj (L ϕ(E ,F )+E F +ϕ(E ,S(F )K )
j i | || | j i j i j i j i
= (−1)Ei Fj ϕ(E ,F )(L −K )+(−1)Ei Fj E F ,
| || | i j j i | || | i j
[E ,F ]= ϕ(E ,F )(K −L )= δ (−1)i(s 1−r 1)(K −L ). Equation (5.16).
i j i j i j ij || − − i j
ToproveEquation(5.17),wecanassumeM = N = 2. Bycomparingcoefficientsofv ⊗v in
1 2
theidenticalvectorsR S S (v ⊗v ) = S S R (v ⊗v )weobtains s +rs×s s =
23 12 13 4 3 13 12 23 4 3 23 14 14 23
0. After left multiplication by s 1s 1 this relation becomes
−22 −11
(5.19) E s 1s +s 1s E = 0.
2 −11 14 −11 14 2
10 HUAFENGZHANG
Let us express s 1s ,s 1s in terms of E ,E ,E by using Lemma 5.1:
−11 13 −11 14 1 2 3
rs rs
s 1s = (rs)s 1s 1s s = s 1s 1(s s −s s )= (s 1E E −E E ),
−11 13 −11 −22 13 22 r−s −11 −22 12 23 23 12 r−s − 1 2 2 1
rs rs
s 1s = s 1s 1(s s −s s ) = (r 1s 1s E −E s 1s )
−11 14 s−r −11 −33 13 34 34 13 s−r − −11 13 3 3 −11 13
−(rs)2
= r 1(s 1E E −E E )E −E (s 1E E −E E )
(r−s)2 − − 1 2 2 1 3 3 − 1 2 2 1
(cid:0) (cid:1)
Equation (5.17) comes from Equation (5.19) and E E = E E ,E2 = 0. (cid:3)
1 3 3 1 2
Let U (sl(M,N)) be the subalgebra of U generated by the E ,F ,K 1,L 1 with 1 ≤
r,s i i i± ±i
i < M+N; this is a sub-Hopf-superalgebra. We believe that Proposition 5.2 and Corollary
2.8 exhaust the defining relations of U (sl(M,N)) as in [DF, Theorem 2.1]. A proof
r,s
(Proposition 6.3) will be given for M = N = 2, in which case we have: for 1≤ i,j ≤ 3
(5.20) K E K 1 = c E , K F K 1 = c 1F , L F L 1 = d F , L E L 1 = d 1E ,
j i j− ij i j i j− −ij i j i −j ij i j i −j −ij i
r 1s s 1 1 r 1s r 1
− − −
where (c ) = r 1 r 1 and (d ) = s 1 1 s are three-by-three
ij − ij −
   
1 s rs 1 1 r 1 rs 1
− − −
matrices with non-zero entries; furthermore ϕ(K ,L) = c 1. 
i j −ij
6. Two-fold quasi-central extension
In this section we show that Relations (5.17)–(5.18) can be dropped, leading to a two-
fold quasi-central extension of U (sl(2,2)) =: Usl. Namely, we will construct a sextuple
r,s
(U,θ,τ,f,P,Q) where U is a Hopf superalgebra, f : U −→ Usl is a morphism of Hopf
superalgebras, θ,τ ∈ Aut(U) are superalgebra automorphisms and P,Q ∈U such that:
(QC1) f is surjective with kernel generated as an ideal by P,Q;
(QC2) Px = θ(x)P and Qx = τ(x)Q for x ∈ U.
Definition 6.1. U = U is the superalgebra generated by e ,f ,k 1,l 1 for 1 ≤ i≤ 3 with
r,s i i i± i±
e ,f being odd and other generators even, and subject to (view E ,F ,K ,L as e ,f ,k ,l )
2 2 i i i i i i i i
Equations (5.20) and (5.11)–(5.16) where M = N = 2 and |1| = |2| = 0,|3| = 1.
U is a Hopf superalgebra with the same coproduct formula as Equation (5.10). Let U+
(resp. U ) be the subalgebra of U generated by the e ,k 1 (resp. the f ,l 1); these are
− i i± i i±
sub-Hopf-superalgebras. By definition e ,f ,k ,l 7→ E ,F ,K ,L extends to a surjective
i i i i i i i i
morphism of Hopf superalgebras f : U −→ Usl. The following lemma is motivated by
Propositions 2.4 and 2.6, whose proof is omitted.
Lemma 6.2. There exists a unique Hopf pairing ϕ: U+×U −→ C such that
−
ϕ(e ,f )= δ (−1)i(s 1−r 1), ϕ(k ,l )= c 1.
i j ij || − − i j −ij
b
As Hopf superalgebras the quantum double is isomorphic to U via multiplication. The
assignment ei 7→ (−b1)|ei|r−1s−1fi, fi 7→ ei, ki 7→ lbi, li 7→ ki extends uniquely to a
Hopf superalgebra isomorphism U −→ Ucop . (cid:3)
r,s s−1,r−1