Multiparameter quantum Pfaffians

0.15 MB·

by  Naihuan Jing

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0 Multiparameter quantum Pfaffians
2

n
Naihuan Jing and Jian Zhang
a
J

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Abstract. ThemultiparameterquantumPfaffianofthe(p,λ)-quantum
2
groupisintroducedandstudiedtogetherwiththequantumdeterminant,

and an identity relating the two invariants is given. Generalization to
]
A the multiparameter hyper-Pfaffian and relationship with the quantum
minors are also considered.
Q
.
h
t
a
m
1. Introduction
[

In the early study of quantum groups, multiparameter quantum groups
1
v and quantum enveloping algebras were considered along the line of one-
8 parameter quantum groups [13, 12, 10]. It was clear that many of their
5
properties are similar to their one-parameter analog, for example, Artin-
4
Schelter-Tate [1] showed that the multiparameter general linear quantum
7
0 group has the same Hilbert function of the polynomial functions in n2 vari-
1. ables under the so-called (p,λ)-condition (see (2.19)). Further results have
0 beenestablished fortwo- andmulti-parameter quantumgroups[11, 2, 4, 8,
7
5] such as the existence of the quantum determinant, which helps to trans-
1
form quantum semigroups into quantum groups. Recently it is known that
:
v
the quantum Pfaffians can be extended to two-parameter quantum groups
i
X as well [7].
r In this paper, we generalize our recent study of quantum determinants
a
and Pfaffians from two-parameter quantum groups to multiparameter cases.
We will adopt the same approach of quadratic algebras [9] to study quan-
tum determinants and quantum Pfaffians, and express them as the scaling
constants ofquantumdifferential forms(cf. [6]). Inparticular, wewillprove
thatthemultiparameter quantum Pfaffian can bedefinedfor amoregeneral
class of multiparameter quantum matrices and prove the identity between
the quantum determinant and quantum Pfaffian, and also establish their
integrality property for quantum groups under the (p,λ)-conditions.
MSC(2010): Primary: 17B37;Secondary: 58A17,15A75, 15B33,15A15.
Keywords: multiparameterquantum groups,q-determinants,q-Pfaffians,q-hyper-Pfaffians
SupportedbyNSFC(11531004), Fapesp(2015/05927-0) andHumboldtFoundation.
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2 NAIHUANJINGANDJIANZHANG
We also formulate the multiparameter quantum determinants in terms
ofthequasideterminantofthegeneratingmatrix. Generalizingtheone-form
and two-form, we obtain higher degree analogs of the multiparameter Pfaf-
fians and establish the transformation rule of the multiparameter quantum
hyper-Pfaffian under the quantum determinant, which extends some of the
transformation rules of Pfaffians in linear algebra.
2. Quantum determinants
2.1. Quantum semigroup A. Let p = (p ),q = (q ) be two sets of
ij ij
n2 parameters in the complex field C arranged in matrix forms satisfying
the following relations:
p p = 1, p = 1; q q = 1, q = 1.
ij ji ii ij ji ii
For a scalar v, the v-commutator [x,y] is defined by
v
(2.1) [x,y] = xy−vyx.
v
Therefore two elements x and y are q-commutative if [x,y] = 0 for a
qij
parameter q .
ij
We define the unital algebra A as an associative complex algebra gener-
ated by a , 1 6 i,j 6 n subject to the following relations:
ij
(2.2) a a = q a a ,
ik il kl il ik
(2.3) a a = p a a ,
ik jk ij jk ik
q 1
kl
(2.4) a a − a a = q a a − a a ,
ik jl jl ik kl il jk jk il
q q
ij ij
p 1
ij
(2.5) a a − a a = p a a − a a ,
ik jl jl ik ij jk il il jk
p p
kl kl
where i < j and k < l. These can be paraphrased as that the quantum
matrix A = (a ) is row q-commutative, column p-commutative, and sat-
ij
isfies the equality between the q-commutator (resp. p-commutator) of the
diagonal entries and the q-commutator (resp. p-commutator) of the skew
diagonal entries up to a q-factor (resp. p-factor).
The algebra A has a bialgebra structure under the comultiplication
A−→ A⊗A given by
(2.6) ∆(aij) = Xaik ⊗akj,
k
and the counit given by ε(a )= δ , the Kronecker symbol.
ij ij
For any permutation σ in S , the q-inversion associated to the parame-
n
ters q is defined as
ij
(2.7) (−q)σ = (−1)l(σ) Y qσjσi,
i<j
σi>σj
where l(σ) = |{(i,j)|i < j,σ > σ }| is the classical inversion number of σ.
i j
MULTIPARAMETER QUANTUM PFAFFIANS 3
The v-based quantum number is defined to be:
(2.8) [n] = 1+v+···+vn−1,
v
and the quantum factorial [n] ! = [1] [2] ···[n] .
v v v v
LetA = (a )bethematrixwithentries satisfying (2.2)-(2.5). Wedefine
ij
the quantum row-determinant and column-determinant of A as follows.
(2.9) rdet(A) = X(−q)σa1,σ1···an,σn,
σ∈Sn
(2.10) cdet(A) = X(−p)σaσ1,1···aσn,n.
σ∈Sn
The first property we show is that both are group-like elements:
∆(rdet(A)) = rdet(A)⊗rdet(A),
∆(cdet(A)) = cdet(A)⊗cdet(A).
To do this we introduce two copies of commuting quantum exterior al-
gebras associated to the parameters p and q respectively. The first one
ij ij
is
Λ (x) = Chx ,...,x i/I
q 1 n
whereI istheideal x2,q x x +x x |1 6 i < j 6 n andonewritesx∧x′ =
(cid:0) i ij i j j i (cid:1)
x⊗x′ mod I. Then we have that
(2.11) x ∧x = −q x ∧x ,
j i ij i j
(2.12) x ∧x = 0,
i i
where i < j. Then for σ ∈ S ,
n
(2.13) x ∧···∧x = (−q) x ∧···∧x .
σ1 σn σ 1 n
Clearly Λ is a left A-comodule with the coaction µ : Λ → A⊗Λ given
q q q q
by
(2.14) µq(xi) =Xaij ⊗xj.
The row determinant can be computed via the coaction:
(2.15) µ (x ∧···∧x ) = rdet(A)⊗x ∧···∧x .
q 1 n 1 n
Subsequently the comodule structure map (id⊗µ )µ = (∆⊗id)µ implies
q q q
that ∆(rdet(A)) = rdet(A)⊗rdet(A).
Let Λ = Λ (y) be the unital associative algebra Chy ,...,y i/J, where
p p 1 n
J is the ideal (y2,p y y +y y |1 6 i < j 6 n). Using similar convention for
i ij i j j i
x ’s, the relations are
i
(2.16) y ∧y =−p y ∧y ,
j i ij i j
(2.17) y ∧y = 0,
i i
4 NAIHUANJINGANDJIANZHANG
where 1 6 i < j 6 n. The space Λ is a right A-comodule with coaction
p
µ′ : Λ → ∆ ⊗A given by
p p p
n
(2.18) µ′p(yi)= Xyj ⊗aji.
j=1
Similar to rdet one has that ∆(cdet(A)) = cdet(A)⊗cdet(A). In general,
rdet6= cdet. However we will consider a special case while the two determi-
nants are equal.
From now on we assume that (p ,q ) live on the parabola
ij ij
(2.19) p q = λ, λ 6= −1, i< j.
ij ij
These relations (2.19) are referred as the (p,λ) conditions [1]. In the follow-
ing whenever we talk about the multiparameter quantum groups we always
consider those satisfying the (p,λ) relations.
Theorem 2.1. In the bialgebra A with the (p,λ) relations, one has that
rdet(A) = cdet(A).
Proof. Consider the following special linear element Φ in A⊗Λ ⊗Λ :
q p
n
(2.20) Φ = X aij ⊗yi⊗xj = yTAx,
i,j=1
where we have set x = (x ,...,x )T and y = (y ,...,y )T.
1 n 1 n
Put further δ = (δ ,...,δ )T, ∂ = (∂ ,...,∂ )T, and consider the fol-
1 n 1 n
lowing elements:
(2.21) δ = Ax,
(2.22) ∂ = ATy.
n
Let ω = y δ = y a x . It follows from the relations (2.2)-(2.5)
i i i Pj=1 i ij j
and the commutation relations of Λ and Λ that
q p
(2.23) ω ∧ω = 0, 1 6 i6 n,
i i
(2.24) ω ∧ω = λω ∧w , 1 6 i < j 6 n.
j i i j
It follows from (2.23)-(2.24) that
∧nΦ = (X λl(σ))ω1∧···∧ωn
σ∈Sn
= [n] !(x ∧···∧x )(∂ ∧···∧∂ )
λ 1 n 1 n
= [n] !rdet(A)(y ∧···∧y )(x ∧···∧x ).
λ 1 n 1 n
NotethatonecanalsorewriteΦ = n ω′ withω′ = ∂ x = n a y x .
Pi=1 i i i i Pj=1 ji j i
Then it follows that
∧nΦ = [n] !cdet(A)(y ∧···∧y )(x ∧···∧x ),
λ 1 n 1 n
MULTIPARAMETER QUANTUM PFAFFIANS 5
which implies that
rdet(A) = cdet(A).
(cid:3)
Due to this identity, from now on, we will define the multiparameter
quantum determinant for the (p,λ)-quantum group as
detq(A) = X(−q)σa1,σ1···an,σn
(2.25) σ∈Sn
= X(−p)σaσ1,1···aσn,n.
σ∈Sn
For a pair of t indices i ,...,i and j ,...,j , we define the quantum
1 t 1 t
row-minor det (Ai1...it) as in (2.25). Like the determinant, the quantum
q j1...jt
row minor row also equals to the quantum column minor for any pairs of
ordered indices 1 6 i < ··· < i 6 n and 1 6 j < ··· < j 6 n, which
1 t 1 t
justifies the notation.
For any t indices i ,...,i
1 t
(2.26) δi1 ∧···∧δit = X detq(Aij11......ijtt)xj1 ∧···∧xjt,
j1<···<jt
where the sum runs through all indices 1 6 j < ··· < j 6 n. This implies
1 t
that det (Ai1...it) = 0 whenever there are two identical rows.
q j1...jt
As δ ’s obey the wedge relations (2.11)-(2.12), for any t-shuffle σ ∈ S :
i n
16 σ < ··· < σ ,σ < ··· < σ 6n, one has that
1 t t+1 n
δ ∧···∧δ ∧δ ∧···∧δ = (−q) δ ∧···∧δ .
σ1 σt σt+1 σn σ 1 n
Note that x ’s also satisfy the same wedge relations. This then implies
j
the following Laplace expansion by invoking (2.26): for each fixed t-shuffle
σ < ··· <σ ,σ < ··· < σ , one has that
1 t t+1 n
(−q)
(2.27) detq(A) = X (−q)αdetq(Aσα11......σαtt)detq(Aσαtt++11......σαnn),
σ
α
where the sum runs through all t-shuffles α ∈ S such that α < ··· <
n 1
α ,α < ··· < α .
t t+1 n
In particular, for fixed i,k
n (−q )
δikdetq(A) = XQl<j(−qlj)aijdetq(Aˆkjˆ)
j=1 Ql<i li
(2.28)
n (−q )
= XQl>j(−qjl)detq(Aˆkjˆ)aij,
j=1 Ql>i il
whereˆi means the indices 1,...,i−1,i+1,...,n for brevity.
6 NAIHUANJINGANDJIANZHANG
As for the quantum (column) determinant or column-minor, the corre-
sponding Laplace expansion for a fixed r-shuffle (τ ...τ ) of n such that
1 n
τ < ··· < τ ,τ < ··· < τ
1 r r+1 n
(−p)
(2.29) detq(A) = X (−p)βdetq(Aβτ11......τβrr)detq(Aβτrr++11......τβnn),
τ
β
where the sum runs through all r-shuffles β ∈ S such that β < ··· <
n 1
β ,β < ··· < β .
r r+1 n
In particular, we have that for fixed i,k
n (−p )
Ql<j lj ˆj
δikdetq(A) = X (−p )ajidetq(Akˆ)
j=1 Ql<i li
n (−p )
Ql>j jl ˆj
(2.30) = X (−p )detq(Akˆ)aji.
j=1 Ql>i il
Theorem 2.2. In the bialgebra A one has that
n
q
a det (A) = λj−iQl=1 lidet (A)a .
ij q n q ij
q
Ql=1 lj
Proof. Let X = (x ),A′ = (a′ ),A′′ = (a′′) be the matrices with
ij ij ij
entries in A defined by
ˆj
(2.31) x = det (A ),
ij q ˆi
(−q )
(2.32) a′ = Ql<j lj a ,
ij (−q ) ij
Ql<i li
(−p )
(2.33) a′′ = Ql>i il a .
ij (−p ) ij
Ql>j jl
It follows from the Laplace expansion that
A′det = A′XA′′ = det A′′.
q q
Therefore Ql<j(−qlj)a det (A) = Ql>i(−pil)det (A)a . This is exactly
Ql<i(−qli) ij q Ql>j(−pjl) q ij
n
q
(2.34) a det (A) = λj−iQl=1 lidet (A)a .
ij q n q ij
q
Ql=1 lj
(cid:3)
Remark 2.3. [2]det (A)iscentralifandonlyifλj−i n q = n q
q Ql=1 li Ql=1 lj
for any i,j.
Theorem 2.2 implies that det (A) is a regular element in the bialgebra
q
A, therefore we can define the localization A[det −1], which will be denoted
q
as GL (n). In fact, Theorem 2.2 gives the following identity:
p,λ
MULTIPARAMETER QUANTUM PFAFFIANS 7
n
q
(2.35) det (A)−1a = λj−iQl=1 lia det (A)−1.
q ij n ij q
q
Ql=1 lj
By defining the antipode
−q
S(a )= Ql<i lidet (Aˆj)det (A)−1
ij −q q ˆi q
Ql<j lj
(2.36)
−p
= Ql>j jldet (A)−1det (Aˆj)
−p q q ˆi
Ql>i il
the bialgebra A[det −1] becomes a Hopf algebra, thus a quantum group in
q
the sense of Drinfeld.
In fact, the second equation follows from (2.35). Therefore, AS(A) =
S(A)A = I by the Laplace expansions. Subsequently
(id⊗S)∆ = (S ⊗id)∆ = ε.
3. Quasideterminants
Inthissection wewillwork withtheringof fractions ofnoncommutative
elements. First of all let us recall some basic facts about quasideterminants.
Let X be the set of n2 elements x ,1 ≤ i,j ≤ n. For convenience, we also
ij
use X to denote the matrix (x ) over the ring generated by x .
ij ij
Denote byF(X)thefreedivisionringgenerated by0,1,x ,1 ≤ i,j ≤ n.
ij
Itiswell-knownthatthematrixX = (x )isaninvertibleelementoverF(X)
ij
[3] .
Let I,J be two finite subsets of cardinality k ≤ n inside {1,...,n}.
Following [3], we introduce the notion of quasiderminant.
Definition 3.1. For i ∈I,j ∈ J, the (i,j)-th quasideterminant |X| is
ij
the following element of F(X):
|X| = y−1,
ij ji
where Y = X−1 = (y ).
ij
If n = 1, I = i,J = j. Then |X| = x .
ij ij
When n ≥ 2, and let Xij be the (n − 1) × (n − 1)-matrix obtained
from X by deleting the ith row and jth column. In general Xi1···ir,j1···jr
denotes the submatrix obtained from X by deleting the i ,··· ,i -th rows,
1 r
and i ,··· ,i -th columns. Then
1 r
|X|ij = xij −Xxii′(|Xij|j′i′)xj′j,
i′,j′
where the sum runs over i′ ∈/ I \{i},j′ ∈/ J \{j}.
Theorem 3.2. Let A be the matrix of generators of GL (n). In the
p,λ
ring of fractions of elements of GL (n), one has that
p,λ
(3.1) det (A) = |A| |A11| |A12,12| ···a
q 11 22 33 nn
8 NAIHUANJINGANDJIANZHANG
and the quasi-minors in the right-hand side commute with each other. More
generally, for two permutations σ and τ of S , one has that
n
(−q)
(3.2) det (A) = τ|A| |Aσ1τ1| |Aσ1σ2,τ1τ2| ···a .
q (−q) σ1τ1 σ2τ2 σ3τ3 σnτn
σ
Proof. By definition the quasi-determinants of A are inverses of the
entries of the antipode S(A),
(−q )
|A| = S(a )−1 = Ql<i li det (A)det (Aij)−1
ij ji q q
(−q )
Ql<j lj
(3.3)
(−p )
= Ql>j jl det (Aij)−1det (A),
q q
(−p )
Ql>i il
then
(−q )
Ql<j lj |A| det (Aij) = det (A)
ij q q
(−q )
Ql<i li
(3.4)
(−p )
Ql>i il det (Aij)|A| = det (A),
q ij q
(−p )
Ql>j jl
By induction on the size of the matrix A, one sees that (3.1) and (3.2) hold.
It follows from (2.34) that det (A{1,...,s}{1,...,s}) and det (A{1,...,t}{1,...,t})
q q
commute for 1 ≤ s,t ≤ n−1. Any factor on the right hand side of (3.1) can
beexpressedasdet (A{1,...,s}{1,...,s})det (A{1,...,s+1}{1,...,s+1})−1multipliedby
q q
a scalar, therefore they commute with each other. (cid:3)
4. Multiparameter quantum Pfaffians
Definition 4.1. Let B = (b ) be an 2n×2n square p-antisymmetric
ij
matrix with noncommutative entries such that b = −p b ,i < j. The
ji ij ij
multiparameter quantum q-Pfaffian is defined by
Pfq(B) = X(−q)σbσ(1)σ(2)bσ(3)σ(4)···bσ(2n−1)σ(2n),
σ∈Π
where p = (p ),q = (q ),i < j, and the sum runs through the set Π of
ij ij
permutations σ of 2n such that σ(2i−1) < σ(2i),i = 1,...,n.
Notethattheparametersq andp satisfythe(p,λ)condition: p q =
ij ij ij ij
λ.
Proposition 4.2. For any 0 ≤ t ≤ n,
(4.1) Pfq(B)= Xinv(I,Ic)Pfq(BI)Pfq(BIc),
I
MULTIPARAMETER QUANTUM PFAFFIANS 9
where the sum is taken over all subsets I = {i ···i |i < ··· < i } of
1 2t 1 2t
[1,2n], and
(4.2) inv(I,J) = Y (−qji).
i∈I,j∈J,i>j
Proof. Let Ω = b x x , where x ∈ Λ (x). Then
Pi<j ij i j i q
n
(4.3) ^Ω = Pfq(B)x1∧···∧x2n.
On the other hand,
n
^Ω = Ωt^Ωn−t
= XPf(BI)xIPf(BIc)xJ
(4.4)
I,J
= XPf(BI)Pf(BIc)xIxJ
I,J
It is easy to see that x x vanishes unless J = Ic. Therefore
I J
n
^Ω = XPf(BI)Pf(BIc)xIxIc
I
= Xinv(I,Ic)Pf(BI)Pf(BIc)x1∧···∧x2n.
I
Thus we conclude that
Pf(B)= Xinv(I,Ic)Pf(BI)Pf(BIc).
I
(cid:3)
Theorem 4.3. Let B = (b ) be the p-antisymmetric matrix such
ij 1≤i,j≤2n
that b = −p b ,i < j, and assume that the entries of B commute with
ji ij ij
those of a (p,λ)-matrix A = (a ) . Let C = ATBA. Then
ij 1≤i,j≤2n
(4.5) c = −p c , i < j
ji ij ij
and
(4.6) Pf (C)= det (A)Pf (B).
q q q
10 NAIHUANJINGANDJIANZHANG
Proof. We first check that c also form anti-symmetric matrix. We
ij
compute that
cii = Xakibklali = Xakibklali+aliblkaki
k,l k<l
= X(akiali−pklaliaki)bkl = 0.
k<l
For i< j,
cij = Xakibklalj = X(akibklalj +aliblkakj)
k,l k<l
= X(akialj −pklaliakj)bkl = Xdetq(Akijl)bkl,
k<l k<l
cji = Xakjbklali = X(akjali −pklaljaki)bkl
k,l k<l
= X−pij(akialj −pklaliakj)bkl = −pijXdetq(Akijl)bkl
k<l k<l
= −p c .
ij ij
Consider the element
Ω = xtCx,
where we recall that x = (x ,...,x )t and x ∈ Λ (x). Explicitly we have
1 n i q
that Ω= c x x = (1+λ)c x x , therefore
P1≤i,j≤n ij i j Pi<j ij i j
n
(4.7) ^Ω = (1+λ)nPfq(C)x1∧···∧x2n.
n
On the other hand, let ω = a x . Then
i Pj=1 ij j
ω ω = −q ω ω , i < j,
j i ij i j
ω ω = 0.
i i
As Ax = (ω ,...,ω )t, one has that
1 2n
Ω = xtAtBAx = (Ax)tB(Ax)
= X bijωiωj
1≤i,j≤n
= X(1+λ)bijωiωj.
i<j