aa r X i v : . [ m a t h . QA ] A p r Dynamical quantum determinants and Pfaffians
Naihuan Jing and Jian Zhang
Dedicated to Vyjayanthi Chari in honor of her 60th birthday
Abstract.
We introduce the dynamical quantum Pfaffian on the dy-namical quantum general linear group and prove its fundamental trans-formation identity. Hyper quantum dynamical Pfaffian is also intro-duced and formulas connecting them are given.
1. Introduction
Dynamical quantum groups are important generalization of quantumgroups introduced by Etingof and Varchenko [ ] in connection with theelliptic quantum groups [
5, 6 ]. See the review [ ] for the background andrelated literature as well as comparison with usual quantum groups (see [ ]).The dynamical quantum group is in fact some quantum groupoid, thus alsorelated to the deformation of the Poisson groupoid [
16, 19, 21 ]. In thispaper we essentially follow [ ] to define the dynamical quantum group withsome modification [ ].As discussed in [ ] in general, given an R-matrix one can associate cer-tain quantum semigroup A ( R ) via the RT T formulation. Let V be thecomplex n -dimensional vector space with basis v i and dual basis λ i for V ∗ .We consider the dynamical R-matrix R ( λ ) defined on V ⊗ V as follows. R ( λ ) = q n X i =1 e ii ⊗ e ii + n X i
22, 17 ]. For a unified treatment using Manin’s qua-dratic algebras, see [
11, 13 ].Correspondingly on the dynamical quantum general linear semigroupoid F R ( M ( n )), we can also introduce the dynamical quantum determinant, mi-nors and prove that they also enjoy similar favorable properties [
9, 14, 15 ].It turns out that the quantum dynamical determinant is also a central group-like element, and the Laplace expansions for quantum dynamical minors alsoare satisfied in a manner similar to the non-dynamical quantum situation.In particular the quantum dynamical determinant also turns F R ( M ( n )) intoa dynamical quantum groupoid [
18, 20 ].The gaol of this paper is to introduce the quantum dynamical Pfaffianand show that it enjoys favorable properties similar to the quantum groupsituation [ ]. Our main technique is to use quadratic algebras or quan-tum de Rham complexes [ ] to study quantum determinants and quantumPfaffians, and express them as the scaling constants of quantum differen-tial forms (cf. [ ]). In particular, we prove that the dynamical quantumPfaffian satisfies the transformation property:(1.2) Pf( ABA t ) = det( A )Pf( B )even though the identity Pf( A ) = p det( A ) no longer holds for the quantumanti-symmetric matrix.The paper is organized as follows. In section two, we introduce the dy-namical quantum general linear group via the generalized quantum Yang-Baxter R-matrix and review the basic information on quantum dynamicalminors and determinants. In section three, we give a factorization formulafor the dynamical quantum determinant in terms of quasi-determinant ofGelfand and Retakh. In section four, we study the dynamical quantum Pfaf-fians using q -forms. In the last section, quantum dynamical hyper-Pfaffiansare given and their fundamental properties and identities are discussed.
2. Dynamical analogue of the quantum algebra M(n).
In this section, we recall some basic facts about dynamical quantumgroups [ ].Let h ∗ be the dual space of the n -dimensional commutative Lie algebra h and we fix a linear basis { e i } of h ∗ , so h ∗ can be identified with C n . For[1 , n ] = { , , . . . , n } , define ω : [1 , n ] → h ∗ by ω ( i ) = e i .Fix a generic q ∈ C × . For λ ∈ h ∗ , the functional q λ : h −→ C is definedas usual by v q λ ( v ) , v ∈ h . We denote by h ( λ ) and g ( λ ) the following YNAMICAL QUANTUM DETERMINANTS AND PFAFFIANS 3 special meromorphic functionals on h : h ( λ ) = q q − λ − q − q − λ − ,g ( λ ) = h ( λ ) h ( − λ ) = ( q − λ − q − )( q − λ − q )( q − λ − . (2.1)Let M h ∗ be the space of meromorphic functionals on h ∗ . In particular,the above f ( λ ), g ( λ ) are elements inside M h ∗ . Let h -algebra F R ( M ( n )) bethe associative algebra generated by the elements t ij , 1 ≤ i, j ≤ n togetherwith two copies of M h ∗ . The elements of the two copies M h ∗ are f ( λ ) = f ( λ , . . . , λ n ) and f ( µ ) = f ( µ , . . . , µ n ), embedded as subalgebras. Here λ i (resp. µ i ) is a function on h . The defining relations of F R ( M ( n )) consist oftwo types. The first group of relations are given by f ( λ ) f ( µ ) = f ( µ ) f ( λ ) ,f ( λ ) t ij = t ij f ( λ + ω ( i )) ,f ( µ ) t ij = t ij f ( µ + ω ( j )) , (2.2)where f, f , f ∈ M h ∗ . The second set of relations are h ( µ k − µ l ) t ik t il = t il t ik , k < lh ( λ j − λ i ) t jk t ik = t ik t jk , i < jt ik t jl = t jl t ik + ( h ( λ j − λ i ) − h ( µ k − µ l )) t jk t il , i < j, k < lg ( µ k − µ l ) t ik t jl = g ( λ i − λ j ) t jl t ik + ( h ( µ l − µ k ) − h ( λ i − λ j )) t il t jk , i < j, k < l. (2.3)The algebra F R ( M ( n )) has a bigradation defined as follows. Let deg ( t ij ) =( ω ( i ) , ω ( j )) = ( e i , e j ) ∈ N n × N n and extend this multiplicatively. Then F R ( M ( n )) = M ( m,p ) ∈ N n × N n ∪ (0 , F m,p where the summand f ( λ ) , f ( µ ) ∈ F = M ⊗ h , and F m,p = { t | deg ( t ) =( m, p ) ∈ N n × N n } , and the moment maps are given by µ l ( f ) = f ( λ ) , µ r ( f ) = f ( µ ).For α ∈ h ∗ we denote by T α : M h ∗ → M h ∗ the automorphism ( T α f )( λ ) = f ( λ + α ) for all λ ∈ h ∗ . The algebra F R ( M ( n )) has a comultiplication∆ : F R ( M ( n )) −→ F R ( M ( n )) ⊗ F R ( M ( n )) given by∆( t ij ) = X k t ik ⊗ t kj , ∆( f ( λ )) = f ( λ ) ⊗ , ∆( f ( µ )) = 1 ⊗ f ( µ ) . (2.4)and the counit ε given by ε ( t ij ) = δ ij T − ω ( i ) , ε ( f ( λ )) = ε ( f ( µ )) = f andthe map is extended as a homomorphism and we equip F R ( M ( n )) with thestructure of an h -bialgebroid. NAIHUAN JING AND JIAN ZHANG
Definition . An h -space V is a vector space over M h ∗ equippedwith a diagonalizable h -module, i.e. V = P α ∈ h ∗ V α , with M h ∗ V α ∈ V α forall α ∈ h ∗ . A morphism of h -spaces is an h -invariant h ∗ -linear map.We next define the tensor product of an h -bialgebroid A and an h -space V . Put V e ⊗A = L α,β ∈ h ∗ ( V α ⊗ h ∗ A αβ ) where ⊗ h ∗ denotes the usual tensorproduct modulo the relations v ⊗ µ l ( f ) a = f v ⊗ a . The grading V α ⊗ h ∗ A αβ ⊆ ( V ⊗ A ) β for all α and f ( v ⊗ a ) = v ⊗ µ r ( f ) a makes V e ⊗A into an h -space.Analogously A e ⊗ V = L α,β ∈ h ∗ ( A αβ ⊗ h ∗ V β ) where ⊗ h ∗ denotes the usualtensor product modulo the relations µ r ( f ) a ⊗ v = a ⊗ f v . The grading A αβ ⊗ h ∗ V β ⊆ ( A ⊗ V ) α and f ( a ⊗ v ) = µ l ( f ) a ⊗ v makes A e ⊗ V into an h -space.We now construct two special F R ( M ( n ))-comodules. Let W = M h h w i i be the unital associative algebra generated by the elements w i , ≤ i ≤ n and M h ∗ , its elements denoted by f ( λ ), subject to the relations w i = 0 , ≤ i ≤ nw j w i = − h ( λ j − λ i ) w i w j , ≤ i < j ≤ n, (2.5)as well as the relation f ( λ ) w i = w i f ( λ + ω ( i )) for all f ∈ M h ∗ .Let V = M h h v i i be the unital associative algebra generated by the el-ements v i , ≤ i ≤ n and a copy of M h ∗ embedded as a subalgebra, itselements denoted by f ( λ ) subject to the relations v i = 0 , ≤ i ≤ nv i v j = − h ( λ i − λ j ) v j v i , ≤ i < j ≤ n, (2.6)plus that f ( λ ) v i = v i f ( λ + ω ( i )) for all f ∈ M h ∗ .The following result is easy to see. Theorem . [ ] Define α R (1) = 1 ⊗ , α R ( w i ) = P nj =1 w j ⊗ t ji , α L (1) = 1 ⊗ , α L ( v i ) = P nj =1 t ij ⊗ v j . Then α L extends uniquely to α L : V → F R ( M ( n )) ⊗ V such that V is a left h -comodule algebra for F R ( M ( n )) and α R extends uniquely to α R : W → W ⊗ F R ( M ( n )) such that W is aright h -comodule algebra for F R ( M ( n )) . Let I be a subset of [1 , n ] with entries i < i < · · · < i r and S r be thesymmetric group in r letters. For an element σ ∈ S r , we use l ( σ ) denote thelength of σ . The generalized sign functions S ( σ, I ) and ˜ S ( σ, I ) are definedas follows: S ( σ, I )( λ ) = Y ≤ k
I, J , J be subsets of { , , . . . , n } . If J = J ∪ J , | I | = | J | . Then µ r (sign( J ; J )) ξ IJ = X I ∪ I = I µ l (sign( I ; I )) ξ I J ξ I J , (2.15) ξ JI = X I ∪ I = I ξ J I µ l (sign( J ; J )) µ r (sign( I ; I )) ξ I J . (2.16) NAIHUAN JING AND JIAN ZHANG
It is easy to see from Proposition 2.4 that for any i, j ∈ [1 , n ] δ ij det = n X k =1 sign( { k } ; ˆ k )( λ )sign( { i } ;ˆ i )( µ ) t kj ξ ˆ k ˆ i ,δ ij det = n X k =1 t jk sign(ˆ i ; { i } )( λ )sign(ˆ k ; { k } )( µ ) ξ ˆ i ˆ k ,δ ij det = n X k =1 sign(ˆ k ; { k } )( λ )sign(ˆ i ; { i } )( µ ) ξ ˆ k ˆ i t kj ,δ ij det = n X k =1 ξ ˆ i ˆ k sign( { i } ;ˆ i )( λ )sign( { k } ; ˆ k )( µ ) t jk . (2.17) Lemma . [ ] In F R ( M ( n )) , the determinant commutes with allquantum minor determinants. In particular, det commutes with all gen-erators t ij . Moreover, ∆(det) = det ⊗ det and ε (det) = T − , with , . . . , ∈ h ∗ . Proposition . [ ] The h -bialgebroid F R ( M ( n )) is an h -Hopf al-gebroid with the antipode S defined on the generators by S (det − ) = det , S ( µ r ( f )) = µ l ( f ) , S ( µ l ( f )) = µ r ( f ) for all f ∈ M h ∗ and S ( t ij ) = det − µ l (sign(ˆ j ; { j } )) µ r (sign(ˆ i ; { i } )) ξ ˆ j ˆ i and extended as an algebra antihomomorphism.
3. Quasideterminants and Dieudonn´e determinants
Throughout this section we work with rings of fractions of noncommu-tative rings.
Definition . [ ] Let X = ( x ij ) be an n × n matrix over a ring withidentity such that its inverse matrix X − exists, and the ( j, i )th entry of X − is an invertible element of the ring. Then the ( ij )th quasideterminant of Xis defined by the formula | X | ij = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x . . . x j . . . x n . . . . . .x i . . . x ij . . . x in . . . . . .x n . . . x nj . . . x nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (X − ) − ji , where the first or the second notation with x ij denotes the quasidetermi-nant.When n ≥
2, and let X ij be the ( n − × ( n − i th row and j th column. In general X i ··· i r ,j ··· j r YNAMICAL QUANTUM DETERMINANTS AND PFAFFIANS 7 denotes the submatrix obtained from X by deleting the i , · · · , i r -th rows,and i , · · · , i r -th columns. Then | X | ij = x ij − X i ′ ,j ′ x ii ′ ( | X ij | j ′ i ′ ) x j ′ j , where the sum runs over i ′ / ∈ I \ { i } , j ′ / ∈ J \ { j } . Theorem . Let T be the matrix of generators t ij of F R ( M ( n )) , σ = i . . . i n and τ = j . . . j n be two permutations of S n . In the ring of fractionsof F R ( M ( n )) , one has that (3.1) det( T ) = Q nk =1 µ l (sign( I kc ; i k )) Q nk =1 µ r (sign( J kc ; j k )) t i n j n . . . | T i j | i j | T | i j where I k = { i , . . . , i k } , J k = { j , . . . , j k } . Proof.
By definition the quasi-determinants of T are inverses of theentries of S ( T ),(3.2) | T | ij = S ( t ji ) − = ξ ˆ i ˆ j − µ r (sign(ˆ j ; { j } )) µ l (sign(ˆ i ; { i } )) det( T ) , then(3.3) det( T ) = µ l (sign(ˆ i ; { i } )) µ r (sign(ˆ j ; { j } )) ξ ˆ i ˆ j | T | ij . Eqs (3.1) follows from induction on n . (cid:3) Remark . If i k = j k = n + 1 − k for any k , all the factors on theright hand side of 3.1 commute with each other. In general the factors donot commute.
4. Dynamical quantum Pfaffians
First we review the general theory of the Pfaffian [
11, 12 ], and weassume the minimum condition here. Let B be the algebra generated bythe elements b ij for 1 ≤ i < j ≤ n , and a copy of M h ∗ embedded as asubalgebra, its elements denoted by f ( λ ). The dynamical quantum Pfaffianis defined byPf( B ) = X σ ∈ Π S ( σ ) b σ (1) σ (2) b σ (3) σ (4) · · · b σ (2 n − σ (2 n ) , where S ( σ ) = S ( σ, [1 , n ]), Π is the set of permutations σ of 2 n such that σ (2 i − < σ (2 i ) , i = 1 , . . . , n. For any two disjoint subsets I , I of [1 , n ], we define the dynamicalquantum sign functions sign( I , I ) and g sign( I , I ) bysign( I , I ) = Y k>l ; k ∈ I ,l ∈ I ( − h ( λ k − λ l )) , g sign( I , I ) = Y k Let I = { i , i , . . . , i k } with 1 ≤ i < i < . . . , i k ≤ n . Denote by B I the submatrix of B with the rows and columns indexed by I . The followingresult gives an iterative algorithm to compute the dynamic Pfaffian. Proposition . For each ≤ t ≤ n we have that (4.2) Pf( B ) = X I sign( I ; I c )Pf( B I )Pf( B I c ) , where the sum runs over all subsets I of [1 , n ] such that | I | =2t and I c isthe complement of I . Proof. Define the tensor product of W ⊗ B to be the usual tensorproduct modulo the relations f w ⊗ b = w ⊗ f b . Let Ω = P i Theorem . Denote by F R ( M (2 n )) e ⊗B the usual tensor product mod-ulo the relations µ r ( f ) t ⊗ b = t ⊗ f b and f ( t ⊗ b ) = µ l ( f ) t ⊗ b . Let ξ ijkl bethe × -dynamical quantum minors in F R ( M (2 n )) , b ij the generators of B , and c ij = P k Let w ⊗ t ⊗ b be an element of W e ⊗ F R ( M ( n )) e ⊗B , then f w ⊗ t ⊗ b = w ⊗ µ l ( f ) t ⊗ b = w ⊗ f ( t ⊗ b ) ,f ( w ⊗ t ) ⊗ b = w ⊗ µ r ( f ) t ⊗ b = w ⊗ t ⊗ f ( b ) . (4.5)Let δ i = P nj =1 w j ⊗ t ji , and consider the element Ω = P i,j w i w j ⊗ c ij .It is clear that(4.6) Ω n = w · · · w n ⊗ Pf( C ) , YNAMICAL QUANTUM DETERMINANTS AND PFAFFIANS 9 where the product is the wedge product among w i . On the other hand,Ω = P δ i δ j ⊗ b ij . Then(4.7) Ω n = δ · · · δ n ⊗ Pf( B ) = w · · · w n ⊗ det( T ) ⊗ Pf( B ) . Comparing (4.13) and (4.14) we conclude thatPf( C ) = det( T ) ⊗ Pf( B ) . (cid:3) Let e B be the algebra generated by the elements e b ji for 1 ≤ i < j ≤ n ,and a copy of M h ∗ embedded as a subalgebra, its elements denoted by f ( λ ).the dynamical quantum Pfaffian is defined by f Pf( e B ) = X σ ∈ Π ˜ S ( σ ) e b σ (2 n ) σ (2 n − e b σ (2 n − σ (2 n − · · · e b σ (2) σ (1) , where Π is the set of shuffle permutations σ of 2 n such that σ (2 i − <σ (2 i ) , i = 1 , . . . , n. For any two disjoint subsets I , I of { , , . . . , n } , we define the dy-namical quantum sign by(4.8) g sign( I , I ) = Y k Proposition . For any ≤ t ≤ n , we have that (4.9) f Pf( e B ) = X I g sign( I ; I c ) f Pf( e B I ) f Pf( e B I c ) , where the sum runs through all subsets I of [1 , n ] such that | I | =2t. Proof. Define the tensor product of e B ⊗ V to be the usual tensor prod-uct modulo the relations f b ⊗ v = b ⊗ f v . Let e Ω = P i>j b ij ⊗ v i v j , then(4.10) n ^ e Ω = f Pf( e B ) ⊗ v n ∧ · · · ∧ v . On the other hand, n ^ e Ω = e Ω t ^ e Ω n − t = X I,J (cid:16)f Pf( e B I ) ⊗ v I (cid:17) (cid:16)f Pf( e B J ) ⊗ v J (cid:17) = X I,J f Pf( e B I ) f Pf( e B J ) ⊗ v I v J . (4.11)It is easy to see that v I v J vanishes unless J = I c . Thus we conclude that f Pf( e B ) = X I g sign( I ; I c )Pf( e B I )Pf( e B I c ) . (cid:3) Theorem . Denotes by e B e ⊗ F R ( M (2 n )) the usual tensor product mod-ulo the relations b ⊗ µ l ( f ) t = f b ⊗ t and f ( b ⊗ t ) = b ⊗ µ r ( f ) t . Let ξ klij bethe dynamical quantum minor in F R ( M ( n )) , c ji = P k Let b ⊗ t ⊗ v be an element of B e ⊗ F R ( M ( n )) e ⊗ V , then b ⊗ t ⊗ f v = b ⊗ µ r ( f ) t ⊗ v = f ( b ⊗ t ) ⊗ v,b ⊗ f ( t ⊗ v ) = b ⊗ µ l ( f ) t ⊗ v = f ( b ) ⊗ t ⊗ v. (4.12)Let δ i = P nj =1 t ij ⊗ v j , and consider the element e Ω = P c ji ⊗ v j v i . It isclear that(4.13) e Ω n = f Pf( C ) ⊗ v n · · · v . On the other hand, e Ω = P b lk ⊗ δ l δ k . Then(4.14) e Ω n = f Pf( C ) ⊗ δ n · · · δ = f Pf( e B ) ⊗ det( T ) ⊗ δ n · · · δ . Comparing (4.13) and (4.14) we conclude that f Pf( C ) = f Pf( e B ) ⊗ det( T ) . (cid:3) We now generalize the notion of the dynamical quantum Pfaffian to thedynamical quantum hyper-Pfaffian. Let B be the algebra generated by theelements b i ··· i m , i < i < · · · < i m , ≤ i k ≤ mn, k = 1 , . . . , m , and acopy of M h ∗ embedded as a subalgebra, its elements denoted by f ( λ ). Thedynamical quantum hyper-Pfaffian is defined byPf m ( B ) = X σ ∈ Π S ( σ ) b σ (1) ··· σ ( m ) · · · b σ ( m ( n − ··· σ ( mn ) , where Π is the set of permutations σ of mn such that σ (( k − m + 1) <σ (( k − m + 2) < · · · < σ ( km ) , k = 1 , . . . , n. Note that the dynamical hyper-Pfaffian uses only the entries b i ··· i m ,where i < · · · < i m . Clearly Pf ( B ) = Pf( B ), the quantum dynamicalPfaffian. Proposition . For any ≤ t ≤ n , (4.15) Pf m ( B ) = X I sign( I ; I c )Pf m ( B I )Pf m ( B I c ) , where the sum is taken over all subset I of [1 , mn ] such that | I | = mt . Let I = { i , i , · · · , i m } such that i < i < · · · < i m . We denote by b I the element b i ··· i m . YNAMICAL QUANTUM DETERMINANTS AND PFAFFIANS 11 Theorem . Denotes by F R ( M ( mn )) e ⊗B the usual tensor productmodulo the relations µ r ( f ) t ⊗ b = t ⊗ f b and f ( t ⊗ b ) = µ l ( f ) t ⊗ b . Let ξ IJ be the dynamical quantum minor in F R ( M ( n )) , c I = P J ξ IJ ⊗ b J . Then in F R ( M ( mn )) e ⊗B we have Pf m ( C ) = det( T ) ⊗ Pf m ( B ) . Let e B be the algebra generated by the elements e b i ··· i m , i > i > · · · >i m , ≤ i k ≤ mn, k = 1 , . . . , m. Define dynamical quantum hyper-Pfaffianby f Pf m ( e B ) = X σ ∈ Π e S ( σ ) e b σ ( mn ) ··· σ ( mn − m +1) e b σ ( mn − m ) σ ( mn − m +1) · · · e b σ ( m ) ··· σ (1) , where Π is the set of permutations σ of mn such that σ (( k − m + 1) <σ (( k − m + 2) < · · · < σ ( km ) , k = 1 , . . . , n. Clearly f Pf ( e B ) = f Pf( e B ) discussed before. Proposition . For any ≤ t ≤ n , (4.16) f Pf m ( e B ) = X I g sign( I ; I c ) f Pf m ( e B I ) f Pf m ( e B I c ) , where the sum is taken over all subset I of [1 , mn ] such that | I | = mt . Let I = { i , i , · · · , i m } such that i < i < · · · < i m . We denote by e b I the element b i m ··· i . The following result is proved by the similar method asin the case of m = 2. Theorem . Denotes by e B e ⊗ F R ( M ( mn )) the usual tensor productmodulo the relations b ⊗ µ l ( f ) t = f b ⊗ t and f ( b ⊗ t ) = b ⊗ µ r ( f ) t . Let ξ JI be the dynamical quantum minor in F R ( M ( n )) , c I = P J e b J ⊗ ξ JI . 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