Realization of Uq(sp₂n) within the Differential Algebra on Quantum Symplectic Space

We realize the Hopf algebra Uq(sp₂n) as an algebra of quantum differential operators on the quantum symplectic space X(fs;R) and prove that X(fs;R) is a Uq(sp₂n)-module algebra whose irreducible summands are just its homogeneous subspaces. We give a coherence realization for all the positive root ve...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2017
Main Authors:	Zhang, J., Hu, N.
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Language:	English
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Cite this:	Realization of Uq(sp₂n) within the Differential Algebra on Quantum Symplectic Space / J. Zhang, N. Hu // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 25 назв. — англ.

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spelling	Zhang, J. Hu, N. 2019-02-19T19:45:56Z 2019-02-19T19:45:56Z 2017 Realization of Uq(sp₂n) within the Differential Algebra on Quantum Symplectic Space / J. Zhang, N. Hu // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 25 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B10; 17B37; 20G42; 81R50; 81R60; 81T75 DOI:10.3842/SIGMA.2017.084 https://nasplib.isofts.kiev.ua/handle/123456789/149279 We realize the Hopf algebra Uq(sp₂n) as an algebra of quantum differential operators on the quantum symplectic space X(fs;R) and prove that X(fs;R) is a Uq(sp₂n)-module algebra whose irreducible summands are just its homogeneous subspaces. We give a coherence realization for all the positive root vectors under the actions of Lusztig's braid automorphisms of Uq(sp₂n). The authors would appreciate the referees for their useful comments and good suggestions for improving the paper. The first author is supported by the NSFC (Grants No. 11101258 and No. 11371238). The second author is supported by the NSFC (Grant No. 11771142). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Realization of Uq(sp₂n) within the Differential Algebra on Quantum Symplectic Space Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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title	Realization of Uq(sp₂n) within the Differential Algebra on Quantum Symplectic Space
spellingShingle	Realization of Uq(sp₂n) within the Differential Algebra on Quantum Symplectic Space Zhang, J. Hu, N.
title_short	Realization of Uq(sp₂n) within the Differential Algebra on Quantum Symplectic Space
title_full	Realization of Uq(sp₂n) within the Differential Algebra on Quantum Symplectic Space
title_fullStr	Realization of Uq(sp₂n) within the Differential Algebra on Quantum Symplectic Space
title_full_unstemmed	Realization of Uq(sp₂n) within the Differential Algebra on Quantum Symplectic Space
title_sort	realization of uq(sp₂n) within the differential algebra on quantum symplectic space
author	Zhang, J. Hu, N.
author_facet	Zhang, J. Hu, N.
publishDate	2017
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	We realize the Hopf algebra Uq(sp₂n) as an algebra of quantum differential operators on the quantum symplectic space X(fs;R) and prove that X(fs;R) is a Uq(sp₂n)-module algebra whose irreducible summands are just its homogeneous subspaces. We give a coherence realization for all the positive root vectors under the actions of Lusztig's braid automorphisms of Uq(sp₂n).
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/149279
citation_txt	Realization of Uq(sp₂n) within the Differential Algebra on Quantum Symplectic Space / J. Zhang, N. Hu // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 25 назв. — англ.
work_keys_str_mv	AT zhangj realizationofuqsp2nwithinthedifferentialalgebraonquantumsymplecticspace AT hun realizationofuqsp2nwithinthedifferentialalgebraonquantumsymplecticspace
first_indexed	2025-11-25T18:56:17Z
last_indexed	2025-11-25T18:56:17Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 084, 21 pages Realization of Uq(sp2n) within the Differential Algebra on Quantum Symplectic Space Jiao ZHANG † and Naihong HU ‡ † Department of Mathematics, Shanghai University, Baoshan Campus, Shangda Road 99, Shanghai 200444, P.R. China E-mail: zhangjiao@shu.edu.cn ‡ Department of Mathematics, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Minhang Campus, Dong Chuan Road 500, Shanghai 200241, P.R. China E-mail: nhhu@math.ecnu.edu.cn Received April 18, 2017, in final form October 20, 2017; Published online October 27, 2017 https://doi.org/10.3842/SIGMA.2017.084 Abstract. We realize the Hopf algebra Uq(sp2n) as an algebra of quantum differential operators on the quantum symplectic space X (fs; R) and prove that X (fs; R) is a Uq(sp2n)- module algebra whose irreducible summands are just its homogeneous subspaces. We give a coherence realization for all the positive root vectors under the actions of Lusztig’s braid automorphisms of Uq(sp2n). Key words: quantum symplectic group; quantum symplectic space; quantum differential operators; differential calculus; module algebra 2010 Mathematics Subject Classification: 17B10; 17B37; 20G42; 81R50; 81R60; 81T75 1 Introduction Quantum analogues of differential forms and differential operators on quantum groups or Hopf algebras or quantum spaces have been studied extensively since the end of 1980s (see [4, 7, 10, 15, 17, 23, 24], etc. and references therein). As a main theme of noncommutative (differential) geo- metry, the general theory of bicovariant differential calculus on quantum groups or Hopf algebras was established in [24]. Woronowicz’s axiomatic description of bicovariant bimodules (namely, Hopf bimodules in Hopf algebra theory) is not only used to construct/classify the first order differential calculi (FODC) on Hopf algebras, but also leads to the appearance of Woronowicz’s braiding [24, Proposition 3.1] (also see [22, Theorem 6.3]). Actually, the defining condition of Yetter–Drinfeld module appeared implicitly in Woronowicz’s work a bit earlier than [20, 25] (see [24, formula (2.39)]), as was witnessed by Schauenburg in [22, Corollaries 6.4 and 6.5] proving that the category of Woronowicz’s bicovariant bimodules is categorically equivalent to the category of Yetter–Drinfeld modules, while the latter has currently served as an important working framework for classifying the finite-dimensional pointed Hopf algebras. The coupled pair consists of a quantum group and its corresponding quantum space on which it coacts, both of which in the pair were intimately interrelated [21]. On the other hand, the covariant differential calculus on the quantum space Cnq was built by Wess–Zumino [23] so as to extend the covariant coaction of the quantum group GLq(n) to quantum derivatives. Along the way, many pioneering works appeared by Ogievetsky et al. [17, 18, 19], etc. Recall that for any bialgebra A, by a quantum space for A we mean a right A-comodule algebra X . Here, we let A denote a certain Hopf quotient of the FRT bialgebra A(R), which is related with a standard R-matrix R of the ABCD series (cf. [10, 21]), and we set X := Xr(fs; R) mailto:zhangjiao@shu.edu.cn mailto:nhhu@math.ecnu.edu.cn https://doi.org/10.3842/SIGMA.2017.084 2 J. Zhang and N. Hu (adopting the notation in the book [10]). For the definition of polynomials fs in types ABCD, we refer to [10, Definitions 4, 8, 12 in Sections 9.2 and 9.3]. Roughly speaking, viewing Uq(g) as the Hopf dual object of quantum group Gq in types ABCD, one sees that the aforementioned quantum space X is a left Uq(g)-module algebra. As a benefit of the viewpoint, this allows one to do the crossed product construction to enlarge the quantum enveloping algebra Uq(g) into a quantum enveloping parabolic subalgebra of the same type but with a higher rank. This actually contributes an evidence to support Majid’s conjecture [14] on the rank-inductive construction of Uq(g)’s via his double-bosonization procedure (see also a recent work [5] for confirming Majid’s claim in the classical cases). For types B and D, under the assumption that q is not a root of unity, Fiore [2] used the standard R-matrix for the quantum group SOq(N) (N = 2n+ 1 or 2n) to define some quantum differential operators on the quantum Euclidean space RNq . Then he realized Uq−1(soN ) within the differential algebra Diff(RNq ) such that RNq is a left Uq−1(soN )-module algebra, and further developed the corresponding quantum Euclidean geometry in his subsequent works. There were many works [17, 18, 19], prior to [2], using quantum differential operators to describe the GLq(n) and SOq(n), q-Lorentz algebra, and q-deformed Poincaré algebra, etc. For type A, there appeared several special discussions in rank 1 case, see [9, 16, 23], etc. To our interest, for arbitrary rank, different from [17] and [2], the second author [6] intro- duced the notion of quantum divided power algebra Aq(n) for q both generic and root of unity. He defined q-derivatives over Aq(n) and realized the U -module algebra structure of Aq(n) for U = Uq(sln), uq(sln). A coherence realization of all the positive root vectors in terms of the quantum differential operators was provided (in the modified q-Weyl algebra Wq(2n)) which are compatible with the actions of Lusztig’s braid automorphisms [13]. Especially, this dis- cussion of q-derivatives resulted in the definition of the quantum universal enveloping algebras of abelian Lie algebras for the first time, and even the new Hopf algebra structure so-called the n-rank Taft algebra (see [7, 11]) in root of unity case. Based on the realization in [6], Gu and Hu [3] gave further explicit results of the module structures on the quantum Grassmann algebra defined over the quantum divided power algebra, the quantum de Rham complexes and their cohomological modules, as well as the descriptions of the Loewy filtrations of a class of interesting indecomposable modules for Lusztig’s small quantum group uq(sln). For type C, it seems lack of corresponding discussions over the quantum symplectic space in the literature. Here we consider the quantum enveloping algebra Uq(sp2n) with n ≥ 2 and its corresponding quantum symplectic space X (fs; R). We assume that q is not a root of unity. We define the q-analogues ∂i := ∂q/∂xi of the classical partial derivatives and introduce left- and right-multiplication operators xiL and xiR as in [9]. Our discussion also does not use theR-matrix as a tool as in [2]. We consider the subalgebra U2n q generated by some quantum differential operators in the quantum differential algebra Diff(X (fs; R)) (we call it the modified q-Weyl algebra of type C, distinctive from the ordinary one, since it contains some extra automorphisms as group-likes inside). Furthermore, we check the Serre relations of U2n q and show X (fs; R) is a Uq(sp2n)-module algebra. At last, we show that the positive root vectors of Uq(sp2n) defined by Lusztig’s braid automorphisms in [13] can be realized precisely by means of the quantum differential operators defined in Section 5. The paper is organized as follows. Section 2 gives the definition of the quantum symplectic space X (fs; R) and derives some useful formulas. In Section 3, we define the quantum differential operators on X (fs; R) and a subalgebra U2n q of Diff(X (fs; R)). We prove that the generators of U2n q satisfy the Serre relations which implies that U2n q is a quotient algebra of Uq(sp2n). We show that X (fs; R) is a Uq(sp2n)-module algebra whose irreducible summands are just its homogeneous subspaces. In Section 4, we provide inductive formulas to calculate all the positive root vectors under the actions of Lusztig’s braid automorphisms of Uq(sp2n) from simple root vectors. In Section 5, we give a coherence realization for all the positive root vectors of Uq(sp2n). Realization of Uq(sp2n) within the Differential Algebra on Quantum Symplectic Space 3 For simplicity, we write X for X (fs; R). Let N0 (resp. N) be the set of nonnegative (resp. positive) integers, R denote the set of real numbers, k the underlying field of characteristic 0. Assume that q is invertible in k and is not a root of unity. Let n ≥ 2 be a positive integer. Set I = {−n,−n+ 1, . . . ,−1, 1, . . . , n− 1, n} and I+ = {1, . . . , n}. 2 Preliminaries 2.1. Recall that the q-number [m]q for m ∈ Z is defined by [m]q := qm−q−m q−q−1 . Note that [0]q = 0. For m ∈ N, the q-factorial is defined by setting [m]q! := [1]q[2]q · · · [m]q and [0]q! := 1. The q-binomial coefficients are defined by[m k ] q := [m]q[m− 1]q · · · [m− k + 1]q [1]q[2]q · · · [k]q for m, k ∈ Z with k > 0, and [ m 0 ] q := 1. So if k > m ≥ 0, then [ m k ] q = 0. Set [A,B]v = AB − vBA for v ∈ k. When v = 1, [·, ·]1 is the commutator [·, ·]. The following three lemmas can be checked directly and will be used many times in Sections 4 and 5. Lemma 2.1. For u, v ∈ k and u 6= 0, if AB = uBA, then [A,BC]v = uB[A,C]v/u, [A,CB]v = [A,C]v/uB, (2.1) [CA,B]v = u[C,B]v/uA, [AC,B]v = A[C,B]v/u. Lemma 2.2. For u, v, w ∈ k and u 6= 0, if AC = uCA, then [[A,B]v, C]w = [A, [B,C]w/u]uv, [[B,A]v, C]w = u[[B,C]w/u, A]v/u. (2.2) Lemma 2.3. We have [A,B]q = −q[B,A]q−1 , [AB,C]q2 = A[B,C]q + q[A,C]qB, (2.3) [[A,B]q, C]q = [A, [B,C]]q2 + [[A,C]q, B]q. (2.4) 2.2. Recall that the simple roots of sp2n are α1 = 2ε1 and αi = εi− εi−1 for 2 ≤ i ≤ n, where εi = (δ1i, . . . , δni) and ε1, . . . , εn form a canonical basis of Rn. Note that here α1 is chosen to be longer than other simple roots. Let ∆+ be the set of positive roots of sp2n, then ∆+ = {2εi,±εl + εk \| 1 ≤ i ≤ n, 1 ≤ l < k ≤ n}. 2.3. Recall that the quantum universal enveloping algebra Uq(sp2n) generated by {Ei, Fi,Ki, K−1i , i ∈ I+} has the defining relations as follows: KiKj = KjKi, KiK −1 i = K−1i Ki = 1, (2.5) KiEjK −1 i = q aij i Ej , KiFjK −1 i = q −aij i Fj , (2.6) [Ei, Fj ] = δij Ki −K−1i qi − q−1i , (2.7) 4 J. Zhang and N. Hu 1−aij∑ t=0 (−1)t [ 1− aij t ] qi EtiEjE 1−aij−t i = 0, i 6= j, (2.8) 1−aij∑ t=0 (−1)t [ 1− aij t ] qi F ti FjF 1−aij−t i = 0, i 6= j, (2.9) where qi = q (αi,αi) 2 , aij = 2(αi,αj) (αi,αi) , and the Cartan matrix (aij) of sp2n in our indices is 2 −1 0 0 · · · · · · 0 −2 2 −1 0 · · · · · · 0 0 −1 2 −1 · · · · · · 0 ... . . . . . . . . . . . . ... ... . . . . . . . . . . . . ... 0 · · · 0 −1 2 −1 0 · · · · · · 0 −1 2  . Note that q1 = q2, qi = q for 1 < i ≤ n. The relations (2.8) and (2.9) are usually called the Serre relations. The algebra Uq(sp2n) is a Hopf algebra equipped with coproduct ∆, counit ε and antipode S defined by ∆(Ei) = Ei ⊗Ki + 1⊗ Ei, ∆(Fi) = Fi ⊗ 1 +K−1i ⊗ Fi, (2.10) ∆ ( K±1i ) = K±1i ⊗K ±1 i , (2.11) ε(Ei) = ε(Fi) = 0, ε ( K±1i ) = 1, (2.12) S(Ei) = −EiK−1i , S(Fi) = −KiFi, S ( K±1i ) = K∓1i , for i ∈ I+. 2.4. Set λ = q − q−1. By [10, Proposition 16 in Section 9.3.4], the quantum symplectic space X is the algebra with generators xi, i ∈ I, and defining relations: xjxi = qxixj , i < j, −i 6= j, (2.13) xix−i = q2x−ixi + q2λΩi+1, i ∈ I+, (2.14) where Ωi := ∑ i≤j≤n qj−ix−jxj for i ∈ I+, and X is a vector space with basis { x a−n −n · · ·xann \| a−n, . . . , an ∈ N0 } . By definition, for 1 ≤ i ≤ n− 1, we have Ωi = x−ixi + qΩi+1. (2.15) From relations (2.13) and (2.14), we can obtain the following identities: Ωixk =  q2xkΩi, −n ≤ k ≤ −i, xkΩi, −i < k < i, q−2xkΩi, i ≤ k ≤ n, (2.16) and ΩiΩj = ΩjΩi, i, j ∈ I+. Realization of Uq(sp2n) within the Differential Algebra on Quantum Symplectic Space 5 Set xa := x a−n −n · · ·xann and a := (a−n, a1−n, . . . , an), where a−n, . . . , an ∈ N0. We call the monomial x ai1 i1 · · ·xaimim whose subscripts are placed in an increasing order a normal monomial. Write εi = (0, . . . , 1, . . . , 0) ∈ R2n with 1 in the i-position and 0 elsewhere. Then a = ∑ i∈I aiεi. Set \|a\| = ∑ i∈I ai. Thus X = ⊕ mXm is an N0-graded algebra with Xm = Spank{xa \| \|a\| = m}. By induction and using relations (2.13)–(2.16), we get xix m −i = q2mxm−ixi + qm+1λ[m]qΩi+1x m−1 −i , xmi x−i = q2mx−ix m i + qm+1λ[m]qΩi+1x m−1 i , for i ∈ I+ and m ∈ N0. Hence, for i ∈ I+, we have x−ix a = −i−1∏ j=−n qaj xa+ε−i , xaxi =  n∏ j=i+1 qaj xa+εi , (2.17) xix a =  i−1∏ j=−n qaj  qa−ixa+εi + −i−1∏ j=−n q−aj  qa−i+1λ[a−i]qΩi+1x a−ε−i , (2.18) xax−i =  n∏ j=1−i qaj  qaixa+ε−i + ( n∏ k=i qak )−i−1∏ j=−n q−2aj  qλ[ai]qΩi+1x a−εi . (2.19) The following lemma will be used later. Lemma 2.4. For i ∈ I+, we have Ωix a = ( −i∏ l=−n q2al ) n∑ j=i qj−i+(a1−j+···+aj−1)xa+ε−j+εj  . (2.20) Proof. We prove this lemma by induction on i from n to 1. From (2.13) and (2.16), we have Ωnx a = q2a−nx a−n −n Ωnx a1−n 1−n · · ·x an n = q2a−nx a−n+1 −n xnx a1−n 1−n · · ·x an n = q2a−nqa1−n+···+an−1xa+ε−n+εn . So the formula (2.20) holds for i = n. Suppose (2.20) holds for i > 1. Then from (2.13), (2.15) and (2.16), we obtain Ωi−1x a = ( 1−i∏ l=−n q2al ) x a−n −n · · ·x a1−i 1−i Ωi−1x a2−i 2−i · · ·x an n = ( 1−i∏ l=−n q2al ) x a−n −n · · ·x a1−i 1−i (x1−ixi−1 + qΩi)x a2−i 2−i · · ·x an n = ( 1−i∏ l=−n q2al ) qa2−i+···+ai−2xa+ε1−i+εi−1 + q1+2a1−iΩix a. The induction hypothesis completes the proof. � 3 Quantum differential operators on X (fs; R) 3.1. We define some quantum analogs of differential operators on X . 6 J. Zhang and N. Hu Definition 3.1. For any normal monomial xa and i ∈ I, set ∂i.x a := [ai]qx a−εi , xiL .x a := xix a, xiR .x a := xaxi, µi.x a := qaixa, µ−1i .xa := q−aixa. Let Diff(X ) be the unital algebra of quantum differential operators on X generated by ∂i, xiL , xiR , µi and µ−1i with i ∈ I. This algebra can be described precisely as the smash product of a quantum group Dq and the symplectic space X , where the associative algebra Dq generated by ∂i’s (i ∈ I) as well as µi’s (i ∈ I), acting on X , is a Hopf algebra. For a detailed treatment for type A case, one can refer to [6], where the quantum differential operators algebra is the (modified) quantum Weyl algebra (of type A). Since we only use the actions of these quantum differential operators on X , we omit the explicit presentation of Diff(X ). Since µkµl = µlµk, we write τi := n∏ j=i µj and τ−i := −i∏ j=−n µj for i ∈ I+. Now we define a subalgebra of Diff(X ). Definition 3.2. For i ∈ I+ with i ≥ 2, set e1 := [2]−1q q−1µ−11 ( τ−1−2x−1L + q2τ−12 x−1R ) ∂1, f1 := [2]−1q q−1µ−1−1 ( τ−12 x1R + q2τ−1−2x1L ) ∂−1, k1 := µ2−1µ −2 1 , ei := µi−1µ −1 i τ−1−i−1x−iL∂1−i − τ −1 i xi−1R∂i, fi := −µ1−iµ−1−i τ −1 i+1xiR∂i−1 + τ−1−i x1−iL∂−i, ki := µ−iµ −1 1−iµi−1µ −1 i . Let U2n q be the subalgebra of Diff(X ) generated by {ei, fi, ki, k−1i \| i ∈ I+}. Applying the operators defined in Definition 3.2 to any normal monomial xa, and using Definition 3.1 and (2.17)–(2.19), we get e1.x a = [a1]q2x a+ε−1−ε1 + λ [a1 2 ] q q2−2(a−n+···+a−2)Ω2x a−2ε1 , (3.1) f1.x a = [a−1]q2x a−ε−1+ε1 + λ [a−1 2 ] q q2−2(a−n+···+a−2)Ω2x a−2ε−1 , (3.2) k1.x a = q2(a−1−a1)xa, (3.3) ei.x a = qai−1−ai [a1−i]qx a+ε−i−ε1−i − [ai]qx a+εi−1−εi , (3.4) fi.x a = [a−i]qx a−ε−i+ε1−i − qa1−i−a−i [ai−1]qxa−εi−1+εi , (3.5) ki.x a = qa−i−a1−i+ai−1−aixa, (3.6) for 1 < i ≤ n. The following two lemmas will be used later. Realization of Uq(sp2n) within the Differential Algebra on Quantum Symplectic Space 7 Lemma 3.3. For i, j ∈ I+, we have ei.Ωj =  0, i 6= j, −x−jxj−1, i = j > 1, x2−1, i = j = 1, and fi.Ωj =  0, i 6= j, x1−jxj , i = j > 1, x21, i = j = 1. Proof. It follows immediately from (3.1) and (3.5). � Lemma 3.4. For any two normal monomials xa, xb and i ∈ I+ we have k±1i . ( xaxb ) = ( k±1i .xa )( k±1i .xb ) , ei. ( xaxb ) = ( ei.x a )( ki.x b ) + xa ( ei.x b ) , fi. ( xaxb ) = ( fi.x a ) xb + ( k−1i .xa )( fi.x b ) . Proof. We prove this lemma by induction on \|a\|. For \|a\| = 1, write xa = xj , j ∈ I. The asser- tion of this lemma for \|a\| = 1 can be derived from the relations (2.17) (2.18) (2.20), (3.1)–(3.6) and Lemma 3.3 directly. We omit this straightforward and lengthy verification. Suppose that the lemma holds for any normal monomial xa with \|a\| = m. Let xc be a normal monomial with \|c\| = m+ 1. We can write xc = xjx a, where \|a\| = m and j is the smallest index in (c−n, . . . , cn) such that cj 6= 0. Since xaxb can be written as a linear combination of normal monomials, by the induction hypothesis, we get ki. ( xcxb ) = ki. ( xjx axb ) = (ki.xj) ( ki. ( xaxb )) = (ki.xj) ( ki.x a )( ki.x b ) = ( ki.x c )( ki.x b ) . Then ei. ( xcxb ) = ei. ( xjx axb ) = (ei.xj) ( ki. ( xaxb )) + xj ( ei. ( xaxb )) = (ei.xj) ( ki.x a )( ki.x b ) + xj ( ei.x a )( ki.x b ) + xjx a ( ei.x b ) = ( ei. ( xjx a ))( ki.x b ) + xc ( ei.x b ) = ( ei.x c )( ki.x b ) + xc ( ei.x b ) . Other relations can be proved similarly. � The following lemma can be easily checked by definition. Lemma 3.5. For any m ∈ Z we have [m+ 1]q = q[m]q + q−m = q−1[m]q + qm, (3.7)[ m+ 1 2 ] q − [m 2 ] q = [m]q2 , (3.8)[ m+ 1 2 ] q − q2 [m 2 ] q = q1−m[m]q. (3.9) Now we state one of our main theorems. Theorem 3.6. The generators ei, fi, ki, k −1 i , i ∈ I+, of U2n q satisfy the relations (2.5)–(2.9) after replacing Ei, Fi, Ki, K −1 i by ei, fi, ki, k −1 i , respectively. Hence, there is a unique sur- jective algebra homomorphism Ψ: Uq(sp2n) → U2n q mapping Ei, Fi, Ki, K −1 i to ei, fi, ki, k −1 i , respectively. 8 J. Zhang and N. Hu Proof. The relations (2.5) are clear. Using (3.1)–(3.6), the relations (2.6) can be easily checked. For (2.7), we only prove the case i = j = 1, the others can be checked similarly. For any normal monomial xa, using (3.1)–(3.3) and Lemmas 3.3 and 3.4, we get e1f1.x a = [a−1]q2e1.x a−ε−1+ε1 + λ [a−1 2 ] q q2−2(a−n+···+a−2)e1. ( Ω2x a−2ε−1 ) = [a−1]q2e1.x a−ε−1+ε1 + λ [a−1 2 ] q q2−2(a−n+···+a−2)Ω2 ( e1.x a−2ε−1 ) = [a−1]q2 [a1 + 1]q2x a + λ ([ a1 + 1 2 ] q [a−1]q2 + [a−1 2 ] q [a1]q2 ) q2−2(a−n+···+a−2)Ω2x a−ε−1−ε1 + λ2 [a−1 2 ] q [a1 2 ] q q4−4(a−n+···+a−2)Ω2Ω2x a−2ε−1−2ε1 and f1e1.x a = [a1]q2f1.x a+ε−1−ε1 + λ [a1 2 ] q q2−2(a−n+···+a−2)f1. ( Ω2x a−2ε1) = [a1]q2f1.x a+ε−1−ε1 + λ [a1 2 ] q q2−2(a−n+···+a−2) ( k−11 .Ω2 )( f1.x a−2ε1) = [a1]q2 [a−1 + 1]q2x a + λ ([ a−1 + 1 2 ] q [a1]q2 + [a1 2 ] q [a−1]q2 ) q2−2(a−n+···+a−2)Ω2x a−ε−1−ε1 + λ2 [a−1 2 ] q [a1 2 ] q q4−4(a−n+···+a−2)Ω2Ω2x a−2ε−1−2ε1 . Using (3.7) and (3.8), we obtain [e1, f1].x a = ( [a−1]q2 [a1 + 1]q2 − [a1]q2 [a−1 + 1]q2 ) xa = ( [a−1]q2q 2a1 − [a1]q2q 2a−1 ) xa = [a−1 − a1]q2xa. Since q1 = q2, k1 − k−11 q1 − q−11 .xa = q2(a−1−a1) − q−2(a−1−a1) q2 − q−2 xa = [a−1 − a1]q2xa. Hence [e1, f1] = k1 − k−11 q1 − q−11 . Consider the first Serre relation (2.8). For the case i = 1 and j = 2, we need to prove e21e2 − [2]q2e1e2e1 + e2e 2 1 = 0. Set e1,2 := [e1, e2]q2 . It is equivalent to show [e1, e1,2]q−2 = 0. (3.10) By (3.1), (3.4), (3.6) and Lemmas 3.3–3.5, we get e1,2.x a = −[a1]qq 2+a−1−a2xa+ε−2−ε1 − [a2]qq −2a1xa+ε−1−ε2 − λ[a1]q[a2]qq 3−2(a−n+···+a−2)−a1Ω2x a−ε1−ε2 . Realization of Uq(sp2n) within the Differential Algebra on Quantum Symplectic Space 9 From Lemmas 3.3 and 3.4 and the relation k2e1k −1 2 = q−2e1 which has been proved before, it is easy to show e1,2. ( Ω2x a ) = Ω2 ( e1,2.x a ) + (e1e2.Ω2) ( k1k2.x a ) . Using the above two formulas and the identity q[a1]q[a1 − 1]q2 − [a1]q2 [a1 − 1]q − λ [a1 2 ] q q1−a1 = 0, which is easy to check, we can verify [e1, e1,2]q−2 .xa = 0 by direct computation. So the rela- tion (3.10) holds. Consider the first Serre relation (2.8) for i = 2, j = 1. We need to prove e32e1 − [3]qe 2 2e1e2 + [3]qe2e1e 2 2 − e1e32 = 0. It is equivalent to show [e2, [e1,2, e2]]q2 = 0. (3.11) In order to verify the first Serre relation (2.8) for j = i± 1 and i, j > 1, i.e., e2i ei±1 − [2]qeiei±1ei + ei±1e 2 i = 0, it is equivalent to show [ei, [ei, ei±1]q]q−1 = 0. (3.12) It is not hard to check (3.11) and (3.12) after applying their left hand sides to xa. For \|i−j\| > 2, the first Serre relation is [ei, ej ] = 0, which is obvious. The second Serre relation can be verified similarly. This completes the proof. � Due to the above theorem, we can realize the elements of the quantum group Uq(sp2n) as certain q-differential operators on X . In other words, X is a left Uq(sp2n)-module. Let (H,m, η,∆, ε, S) be a Hopf algebra. Recall that an algebra A is called a left H-module algebra if A is a left H-module, and the multiplication map and the unit map of A are left H-module homomorphisms, that is, h.1A = ε(h)1A, (3.13) h.(a′a′′) = ∑ (h(1).a ′)(h(2).a ′′), (3.14) for any h ∈ H, a′, a′′ ∈ A, where ∆(h) = ∑ h(1) ⊗ h(2). Theorem 3.7. The algebra X is a left Uq(sp2n)-module algebra. Proof. It is sufficient to check (3.13) and (3.14) on the generators of Uq(sp2n), since ε and ∆ are algebra homomorphisms. The relations (2.12) and (3.1)–(3.6) imply (3.13). Lemma 3.4, (2.10) and (2.11) imply (3.14). � 3.2. We consider the decomposition of X into a direct sum of irreducible Uq(sp2n)-submodu- les. Recall that X = ⊕ m∈N0 Xm, where Xm is the subspace of homogeneous elements of degree m. Proposition 3.8. The vector space Xm is a finite-dimensional irreducible Uq(sp2n)-module with highest weight vector xm−n and highest weight mεn. Proof. The assertion follows at once from the facts that the symmetric powers of the vec- tor representation of sp2n are irreducible and the theory of finite-dimensional representations of Uq(g) is very similar to that of g when q is not a root of unity (see [12]), especially, they have the same character formulas for the irreducible modules. � 10 J. Zhang and N. Hu 4 Positive root vectors of Uq(sp2n) We are going to list all positive root vectors of Uq(sp2n) in U2n q . We first recall some notions. Let mij be equal to 2, 3, 4 when aijaji is equal to 0, 1, 2, respectively, where aij are the entries of the Cartan matrix of sp2n. The braid group B associated with sp2n is the group generated by elements s1, . . . , sn subject to the relations sisjsisj · · · = sjsisjsi · · · , i 6= j, where there are mij s’s on each side. Lusztig introduced actions of braid groups on Uq(g) in [12, 13]. The following two propositions can be found in many books, for example [8, 10, 13], etc. Proposition 4.1. To every i, i ∈ I+, there corresponds an algebra automorphism Ti of Uq(sp2n) which acts on the generators Kj , Ej , Fj as Ti(Kj) = KjK −aij i , Ti(Ei) = −FiK−1i , Ti(Fi) = −KiEi, Ti(Ej) = −aij∑ r=0 (−1)rqriE (−aij−r) i EjE (r) i , for i 6= j, Ti(Fj) = −aij∑ r=0 (−1)rq−ri F (r) i FjF (−aij−r) i , for i 6= j, where E (r) i = Eri /[r]qi ! and F (r) i = F ri /[r]qi !. The mapping si 7→ Ti determines a homomorphism of the braid group B into the group of algebra automorphisms of Uq(sp2n). The operators Ti defined by Proposition 4.1 are Lusztig’s T ′′i,−1 [13, Section 37.1.3]. Proposition 4.2. The operators Ti satisfy the following relations. 1. For i, j ∈ I+ with \|i− j\| > 1, we have Ti(Ej) = Ej , TiTj = TjTi. 2. For 2 ≤ i, j ≤ n with \|i− j\| = 1, we have Ti(Ej) = [Ei, Ej ]q, TiTj(Ei) = Ej , TiTjTi = TjTiTj . 3. For 1 ≤ i 6= j ≤ 2, we have T1(E2) = [E1, E2]q2 , [2]qT2(E1) = [E2, [E2, E1]q2 ], T1T2T1(E2) = E2, T2T1T2(E1) = E1, T1T2T1T2 = T2T1T2T1. The Weyl group W of sp2n generated by reflections w1, . . . , wn (corresponding to the simple roots of sp2n) has the longest element w0 whose reduced expression is w0 = γ1 · · · γn, where γi = wiwi−1 · · ·w1 · · ·wi−1wi (cf. [1]). Write w0 = wi1wi2 · · ·wiN for this reduced expres- sion. Then β1 = αi1 , β2 = wi1(αi2), . . . , βN = wi1wi2 · · ·wiN−1(αiN ) exhaust all positive roots of sp2n. Realization of Uq(sp2n) within the Differential Algebra on Quantum Symplectic Space 11 Definition 4.3. The elements Eβr = Ti1Ti2 · · ·Tir−1(Eir), 1 ≤ r ≤ N, are called positive root vectors of Uq(sp2n) corresponding to the roots βr’s. Set αi,i = 2εi and α±l,k = ±εl + εk for 1 ≤ i ≤ n and 1 ≤ l < k ≤ n. We can list all positive roots in the ordering according to the above reduced expression for the longest element w0 as follows α1,1, α1,2, α2,2, α−1,2, α2,3, α1,3, α3,3, α−1,3, α−2,3, . . . , αn−1,n, αn−2,n, . . . , α1,n, αn,n, α−1,n, α−2,n, . . . , α1−n,n. Write Ei,i = Eαi,i and E±l,k = Eα±l,k . It is clear that E1,1 = E1 and we will check in Corollary 4.5 that Eαi = Ei for all 1 < i ≤ n. Set Tγi = TiTi−1 · · ·T1 · · ·Ti−1Ti, 1 ≤ i ≤ n. By Definition 4.3, all the positive root vectors of Uq(sp2n) associated to the above ordering of ∆+ are as follows, for any 1 < j ≤ n, Ej−1,j = Tγ1 · · ·Tγj−1(Ej), (4.1) Ei,j = Tγ1 · · ·Tγj−1TjTj−1 · · ·Ti+2(Ei+1), for 1 ≤ i < j − 1, (4.2) Ej,j = Tγ1 · · ·Tγj−1TjTj−1 · · ·T2(E1), (4.3) E−i,j = Tγ1 · · ·Tγj−1Tj · · ·T1 · · ·Ti(Ei+1), for 1 ≤ i < j. (4.4) Lemma 4.4. For 1 < i ≤ j ≤ n, we have T1([E2, E1]q2) = E2, (4.5) TγiTi+1(Ei) = [Ei, Ei+1]q, (4.6) Tγj−1TjTγj−1(Ej) = Ej , (4.7) TγjTj+1Tj · · ·Ti = (TjTj+1)(Tj−1Tj) · · · (TiTi+1)Tγi−1(TiTi+1 · · ·Tj+1). (4.8) Proof. It is easy to see from (2.6) and (2.7) that[ [E1, E2]q2 ,K −1 1 ] q2 = 0 and [E2, F1] = 0, then by Proposition 4.1 and relations (2.1) and (2.2), we get T1 ( [E2, E1]q2 ) = [T1(E2), T1(E1)]q2 = [ [E1, E2]q2 ,−F1K −1 1 ] q2 = − [ [E1, E2]q2 , F1 ] K−11 = −[[E1, F1], E2]q2K −1 1 = − [ K1 −K−11 q2 − q−2 , E2 ] q2 K−11 = E2. 12 J. Zhang and N. Hu The relation (4.6) is clear, since for i > 1 we have TγiTi+1(Ei) = TiTγi−1TiTi+1(Ei) = TiTγi−1(Ei+1) = Ti(Ei+1) = [Ei, Ei+1]q. We use induction on j to prove (4.7). For j = 2, this is obvious by Proposition 4.2(3). Now suppose that (4.7) holds for some j with 2 < j < n. Then Proposition 4.2(2) and induction yield TγjTj+1Tγj (Ej+1) = TjTγj−1(TjTj+1Tj)Tγj−1Tj(Ej+1) = TjTγj−1(Tj+1TjTj+1)Tγj−1Tj(Ej+1) = TjTj+1Tγj−1TjTγj−1(Tj+1Tj(Ej+1)) = TjTj+1(Tγj−1TjTγj−1(Ej)) = TjTj+1(Ej) = Ej+1. To prove (4.8), we use induction on j − i. For j − i = 0, we have TγjTj+1Tj = TjTγj−1TjTj+1Tj = TjTγj−1Tj+1TjTj+1 = TjTj+1Tγj−1TjTj+1. Suppose that (4.8) holds for some j − i− 1 > 0. Then by induction, we get TγjTj+1Tj · · ·Ti+1Ti = (TjTj+1)(Tj−1Tj) · · · (Ti+1Ti+2)Tγi(Ti+1Ti+2 · · ·Tj+1)Ti = (TjTj+1)(Tj−1Tj) · · · (Ti+1Ti+2)(TγiTi+1Ti)Ti+2 · · ·Tj+1 = (TjTj+1)(Tj−1Tj) · · · (Ti+1Ti+2)(TiTi+1)Tγi−1(TiTi+1Ti+2 · · ·Tj+1). So (4.8) holds. � Using (4.7) and Proposition 4.2(1), we get the following corollary easily. Corollary 4.5. For any 1 < j ≤ n, we have Tγ1 · · ·Tγj−1Tj · · ·T1 · · ·Tj−1(Ej) = Ej , that is, E1−j,j = Eαj = Ej. Proposition 4.6. The positive root vectors of Uq(sp2n) have the following commutation rela- tions: E1,2 = [E1, E2]q2 , (4.9) E−i,j = [E−i,j−1, Ej ]q, 3 ≤ i+ 2 ≤ j ≤ n, (4.10) Ei,j = [Ei,j−1, Ej ]q, 3 ≤ i+ 2 ≤ j ≤ n, (4.11) Ej−1,j = [Ej−1, Ej−2,j ]q, 3 ≤ j ≤ n, (4.12) Ej,j = [2]−1q [E1,j , E−1,j ], 2 ≤ j ≤ n. (4.13) Proof. Relation (4.9) is clear. For i ≥ 1, by Proposition 4.2 and the identity (4.8), we obtain that Tγi+1Ti+2Ti+1Tγi(Ei+1) = Ti+1Ti+2TγiTi+1Ti+2Tγi(Ei+1) = Ti+1Ti+2TγiTi+1Tγi([Ei+2, Ei+1]q) = [Ti+1TγiTi+2Ti+1(Ei+2), Ti+1Ti+2TγiTi+1Tγi(Ei+1)]q = [Ti+1TγiTi+2Ti+1(Ei+2), Ti+1Ti+2(Ei+1)]q = [Ti+1Tγi(Ei+1), Ei+2]q. So Proposition 4.2, (4.4), (4.8) and the above formula show that for 1 ≤ i < j − 1 E−i,j = Tγ1 · · ·Tγj−1Tj · · ·T1 · · ·Ti(Ei+1) Realization of Uq(sp2n) within the Differential Algebra on Quantum Symplectic Space 13 = (Tγ1 · · ·Tγj−2)(Tγj−1TjTj−1 · · ·Ti+1)Tγi(Ei+1) = (Tγ1 · · ·Tγj−2)(Tj−1Tj) · · · (Ti+2Ti+3)(Ti+1Ti+2)Tγi(Ti+1 · · ·Tj−1Tj)Tγi(Ei+1) = (Tγ1 · · ·Tγj−2)(Tj−1Tj) · · · (Ti+2Ti+3)Ti+1TγiTi+2Ti+1Ti+2Tγi(Ei+1) = (Tγ1 · · ·Tγj−2)(Tj−1Tj) · · · (Ti+2Ti+3)Ti+1TγiTi+1Ti+2Ti+1Tγi(Ei+1) = (Tγ1 · · ·Tγj−2)(Tj−1Tj) · · · (Ti+2Ti+3)Tγi+1Ti+2Ti+1Tγi(Ei+1) = (Tγ1 · · ·Tγj−2)(Tj−1Tj) · · · (Ti+2Ti+3)[Ti+1Tγi(Ei+1), Ei+2]q = (Tγ1 · · ·Tγj−2)[Tj−1 · · ·Ti+2Ti+1Tγi(Ei+1), (Tj−1Tj) · · · (Ti+2Ti+3)(Ei+2)]q = [Tγ1 · · ·Tγj−2Tj−1 · · ·Ti+2Ti+1Tγi(Ei+1), Ej ]q = [E−i,j−1, Ej ]q. Hence the relation (4.10) holds. For j = i+ 2, the relations (4.2) and (4.6) yield Ei,i+2 = Tγ1 · · ·Tγi+1Ti+2(Ei+1) = Tγ1 · · ·Tγi([Ei+1, Ei+2]q) = [Ei,i+1, Ei+2]q. For j > i+ 2, by (4.2), (4.6) and (4.8), we have Ei,j = Tγ1 · · ·Tγj−1TjTj−1 · · ·Ti+2(Ei+1) = Tγ1 · · ·Tγj−2(Tj−1Tj)(Tj−2Tj−1) · · · (Ti+2Ti+3)Tγi+1Ti+2 · · ·Tj−1Tj(Ei+1) = Tγ1 · · ·Tγj−2(Tj−1Tj)(Tj−2Tj−1) · · · (Ti+2Ti+3)Tγi+1Ti+2(Ei+1) = Tγ1 · · ·Tγj−2(Tj−1Tj)(Tj−2Tj−1) · · · (Ti+2Ti+3)([Ei+1, Ei+2]q) = [Tγ1 · · ·Tγj−2Tj−1Tj−2 · · ·Ti+2(Ei+1), Ej ]q = [Ei,j−1, Ej ]q. So the relation (4.11) holds. For j ≥ 3, using the relations (4.1) (4.4) and (4.11), we have Ej−1,j = Tγ1 · · ·Tγj−1(Ej) = Tγ1 · · ·Tγj−2Tj−1Tγj−2Tj−1(Ej) = Tγ1 · · ·Tγj−2Tj−1Tγj−2([Ej−1, Ej ]q) = [Tγ1 · · ·Tγj−2Tj−1Tγj−2(Ej−1), Tγ1 · · ·Tγj−2Tj−1(Ej)]q = [E2−j,j−1, Tγ1 · · ·Tγj−2([Ej−1, Ej ]q)]q = [Ej−1, [Tγ1 · · ·Tγj−2(Ej−1), Ej ]q]q = [Ej−1, [Ej−2,j−1, Ej ]q]q = [Ej−1, Ej−2,j ]q. That is, the relation (4.12) holds. It is easy to check that T1Tγi = TγiT1 for any i ∈ I+, so for j ≥ 2, using (4.2)–(4.5) and Proposition 4.2(3), we obtain Ej,j = Tγ1 · · ·Tγj−1TjTj−1 · · ·T2(E1) = [2]−1q Tγ1 · · ·Tγj−1TjTj−1 · · ·T3 ([ E2, [E2, E1]q2 ]) = [2]−1q [ Tγ1 · · ·Tγj−1TjTj−1 · · ·T3(E2), Tγ1 · · ·Tγj−1TjTj−1 · · ·T3 ( [E2, E1]q2 )] = [2]−1q [ E1,j , Tγ1 · · ·Tγj−1TjTj−1 · · ·T3 ( [E2, E1]q2 )] = [2]−1q [ E1,j , Tγ2 · · ·Tγj−1TjTj−1 · · ·T3T1 ( [E2, E1]q2 )] = [2]−1q [ E1,j , Tγ2 · · ·Tγj−1TjTj−1 · · ·T3(E2) ] = [2]−1q [ E1,j , Tγ2 · · ·Tγj−1TjTj−1 · · ·T3T1T2T1(E2) ] = [2]−1q [ E1,j , T1Tγ2 · · ·Tγj−1TjTj−1 · · ·T3T2T1(E2) ] = [2]−1q [E1,j , E−1,j ]. This proves (4.13). � Remark 4.7. By Proposition 4.6, we can perform a double induction first on i then on j with 1 ≤ i ≤ j ≤ n to obtain all the positive root vectors E±i,j from simple root vectors. 14 J. Zhang and N. Hu 5 Realization of positive root vectors of Uq(sp2n) In order to realize all the positive root vectors of Uq(sp2n) directly and concisely as certain operators in Diff(X ), we introduce some new operators. Definition 5.1. For i ∈ I+, set Λ0 = τn+1 = τ−n−1 := 1, Λ−i := −1∏ j=−i µj , Λi := i∏ j=1 µj , D−i := µiτ −1 −i−1∂−i, Di := τ−11 Λ−1i−1∂i, X−iL := µ−1i µ−ix−iL , XiR := Λ2 ixiR , and Φ0 := 0, Ψn+1 := 0, Φi := i∑ j=1 qj−iΛ2 j−1D−jDj , Ψi := τ2−i n∑ j=i qj−iτ−2−j X−jLXjR , X−iR := qiΛ2 1−i ( µ2iX−iL + λµ2−iΨi+1Di ) . Then we get Ψi = X−iLXiR + qµ2−iΨi+1, (5.1) Φi = Λ2 i−1D−iDi + q−1Φi−1. (5.2) The commutation relations in the following three lemmas will be used frequently in this section. Lemma 5.2. 1. For k, l ∈ I and i ∈ I+, we have Dkµl = qδklµlDk, XiRµk = q−δi,kµkXiR , X−iLµk = q−δ−i,kµkX−iL . 2. For i, j ∈ I+ with i < j, we have [Dj ,Di]q = [D−i,D−j ]q = 0, [XjR ,XiR ]q = [X−iL ,X−jL ]q = 0, [XiR ,Dj ]q = [X−jL ,D−i]q = 0, [Di,XjR ]q = [D−j ,X−iL ]q = 0. 3. For i, j ∈ I+ with i 6= j, we have [Di,D−j ] = [X−iL ,XjR ] = 0, [Di,X−jL ] = [D−i,XjR ] = 0, [Di,D−i]q = [XiR ,X−iL ]q = 0, [X−iL ,Di]q = [D−i,XiR ]q = 0. Realization of Uq(sp2n) within the Differential Algebra on Quantum Symplectic Space 15 4. For i ∈ I+, we have DiXiR = qλ−1 ( q2µ2i − 1 ) , XiRDi = qλ−1 ( µ2i − 1 ) , D−iX−iL = λ−1 ( q2µ2−i − 1 ) , X−iLD−i = λ−1 ( µ2−i − 1 ) . Then [Di,XiR ] = q2µ2i , [Di,XiR ]q2 = q2, [XiR ,Di]q−2 = −1, [D−i,X−iL ] = qµ2−i, [D−i,X−iL ]q2 = q, [X−iL ,D−i]q−2 = −q−1. Proof. Applying both sides of each identity to any normal monomial xa, using (2.17) and Definitions 3.1 and 5.1, we can obtain these commutation relations. � By Definition 5.1, Lemmas 2.1 and 5.2 and (5.2), it is easy to check the following lemma. Lemma 5.3. The operators Φi and Ψi satisfy the following commutation relations. 1. For i ∈ I+ and t, k, l ∈ I with \|t\| < i and \|k\| > i, we have [Ψi, µt] = [Φi, µk] = 0, [Ψi, µk]q−1 = [Φi, µt]q = 0, [Ψi, µ±i]q−1 = [Φi, µ±i]q = 0. 2. For i, j ∈ I+ with i < j, we have [Di,Ψj ]q = [D−i,Ψj ]q−1 = 0, [Ψj ,X−iL ]q−1 = [Ψj ,XiR ]q = 0, [Φi,X−jL ]q−1 = [Φi,XjR ]q = 0. 3. For i, j ∈ I+ with i ≤ j, we have [Φi,Dj ]q−1 = [Φi,D−j ]q = 0, [Ψi,XjRDj+1] = −qj+2−iτ2−iτ −2 −j−1X−j−1LXjR , [Ψi,X−j−1LD−j ] = −qj+1−iτ2−iτ −2 −j−1X−j−1LXjR , so [Ψi,XjRDj+1] = q[Ψi,X−j−1LD−j ]. (5.3) 4. For i ∈ I+ we have [Di,Ψi]q = qX−iL , (5.4) [Ψi,X−iL ]q = 0, [Φi,XiR ]q = q2Λ2 iD−i, [Φi,X−iL ]q−1 = Λ2 i−1µ 2 −iDi, [Φi,X−iL ]q = Λ2 i−1Di − λq−1X−iLΦi−1. 5. For i, k ∈ I+, we have [Dk, [Dk,Ψi]q]q−1 = 0. (5.5) 16 J. Zhang and N. Hu From now on, Lemma 2.1 is frequently used without extra explanation. Lemma 5.4. The operators X−iR for i ∈ I+ satisfy the following commutation relations. 1. For i ∈ I+, k, l ∈ I with \|k\| < i and \|l\| 6= i, we have [X−iR , µk] = [ X−iR , µlµ −1 −l ] = [ X−iR , µiµ −1 −i ] q = 0. (5.6) 2. For i ∈ I+, we have [Di,X−iR ]q = [X−iR ,X−iL ] = 0, (5.7) [X−iR ,XiRDi+1]q = q2X−i−1R − q i+3Λ2 −iµ 2 iX−i−1L , (5.8) [X−iR ,X−i−1LD−i]q = −qi+2Λ2 −iµ 2 iX−i−1L . (5.9) 3. For i, j ∈ I+ with i < j, we have [XiR ,X−jR ] = [X−jR ,X−iL ]q = 0, [Di,X−jR ] = [D−i,X−jR ]q = 0. (5.10) Proof. Using Lemmas 5.2 and 5.3, we can prove this lemma directly. We only show (5.8) for example. For i ∈ I+, by definition of X−iR , Lemma 5.2 and (5.1), we have [X−iR ,XiRDi+1]q = [ qiΛ2 1−i ( µ2iX−iL + λµ2−iΨi+1Di ) ,XiRDi+1 ] q = qiΛ2 1−i [ µ2iX−iL ,XiRDi+1 ] q + qiλΛ2 1−i [ µ2−iΨi+1Di,XiRDi+1 ] q = qiΛ2 1−iµ 2 i [X−iL ,XiRDi+1]q−1 + qiλΛ2 1−iµ 2 −i[Ψi+1Di,XiRDi+1]q = 0 + qiλΛ2 −i [( X−i−1LXi+1R + qµ2−i−1Ψi+2 ) Di,XiRDi+1 ] q = qiλΛ2 −iX−i−1L(Xi+1RDiXiRDi+1 − XiRDi+1Xi+1RDi) + qi+1λΛ2 −1−iΨi+2[Di,XiR ]q2Di+1 = qiλΛ2 −iX−i−1L(DiXiRXi+1RDi+1 − XiRDiDi+1Xi+1R) + qi+3λΛ2 −1−iΨi+2Di+1 = qiλΛ2 −iX−i−1L([Di,XiR ]Xi+1RDi+1 − XiRDi[Di+1,Xi+1R ]) + qi+3λΛ2 −1−iΨi+2Di+1 = qi+3Λ2 −iX−i−1L ( µ2i ( µ2i+1 − 1 ) − ( µ2i − 1 ) µ2i+1 ) + qi+3λΛ2 −1−iΨi+2Di+1 = qi+3Λ2 −i ( µ2i+1 − µ2i ) X−i−1L + qi+3λΛ2 −1−iΨi+2Di+1 = q2X−i−1R − q i+3Λ2 −iµ 2 iX−i−1L , that is, (5.8) holds. � Corollary 5.5. For i, j ∈ I+ with i ≤ j, we have [X−jR , [Dj ,Ψi]q] = 0. (5.11) Proof. For 1 ≤ i ≤ j ≤ n, Lemma 5.2 yields [Dj ,Ψi]q = [ Dj , τ 2 −i n∑ k=i qk−iτ−2−kX−kLXkR ] q = τ2−i n∑ k=i qk−iτ−2−k [Dj ,X−kLXkR ]q = −λτ2−i j−1∑ k=i qk−iτ−2−kX−kLXkRDj + qj−i−1τ2−iτ −2 −j X−jL [Dj ,XjR ]q2 Realization of Uq(sp2n) within the Differential Algebra on Quantum Symplectic Space 17 = −λ j−1∑ k=i qk−iτ2−iτ −2 −kX−kLXkRDj + qj−i+1τ2−iτ −2 −j X−jL , then by (5.6), (5.7) and (5.10), we have [X−jR , [Dj ,Ψi]q] = −λ j−1∑ k=i qk−iτ2−iτ −2 −k [X−jR ,X−kLXkRDj ] + qj−i+1τ2−iτ −2 −j [X−jR ,X−jL ] = −λ j−1∑ k=i qk−iτ2−iτ −2 −k [X−jR ,X−kL ]qXkRDj = 0, that is, (5.11) holds. � We are now in the position to realize all the positive root vectors E±i,j of Uq(sp2n) as e±i,j in Diff(X ). Definition 5.6. For i, j ∈ I+ with i < j, set e−i,j := (−1)i+jq−2(XiRDj − [Dj ,Ψi+1]qD−i), ei,i := [2]−1q τ1τ −1 −1 ( X−iRDi + q−2[Di,Ψ1]qDi ) , ei,j := (−1)j+1τ1τ −1 −1 ( X−iLDj + qi−1X−jR [Φi,X−iL ]q ) . The next lemma says that the operators which realize the simple root vectors of Uq(sp2n) defined in Definition 5.6 coincide with those defined in Definition 3.2. Lemma 5.7. e1,1 = e1 and e1−i,i = ei for 1 < i ≤ n. Proof. From (2.20), it is easy to show that Ωix a = τ−11 τ−1τ 2 −i n∑ j=i qj−i−2τ−2−j X−jLXjR .x a for any normal monomial xa. Then it yields from (2.19) that x−iR = τ1τ −1 −i µ 2 iΛ1−iX−iL + λτ1τ−1 n∑ j=i+1 qj−i−1τ−2−j X−jLXjRDi = τ1τ −1 −i µ 2 iΛ1−iX−iL + λτ1Λ−iτ −1 −i−1Ψi+1Di. Write ei in terms of the new operators defined in Definition 5.1. By (5.4), we get e1 = [2]−1q q−1µ−11 ( τ−1−2x−1L + q2τ−12 x−1R ) ∂1 = [2]−1q τ−1−1 ( q−1X−1L + qµ21X−1L + qλµ2−1Ψ2D1 ) τ1D1 = [2]−1q τ1τ −1 −1 ( q−1X−1L + X−1R ) D1 = [2]−1q τ1τ −1 −1 ( q−2[D1,Ψ1]q + X−1R ) D1 = e1,1, and for i > 1 ei = µi−1µ −1 i τ−1−i−1x−iL∂1−i − τ −1 i xi−1R∂i = q−1X−iLD1−i − q−2Xi−1RDi = q−2([Di,Ψi]qD1−i − Xi−1RDi) = e1−i,i. This completes the proof. � Proposition 5.8. The commutation relations (4.9)–(4.13) remain valid if E is replaced by e. 18 J. Zhang and N. Hu Proof. (1) To prove e1,2 = [e1, e2]q2 , we compute the following four brackets first. By (2.3), (5.1), (5.8), (5.9) and Lemma 5.2, we get [X−1RD1,X1RD2]q2 = X−1R [D1,X1RD2]q + q[X−1R ,X1RD2]qD1 = X−1R [D1,X1R ]q2D2 + q ( q2X−2R − q 4Λ2 −1µ 2 1X−2L ) D1 = q2X−1RD2 + q3X−2RD1 − q5µ2−1µ21X−2LD1 = q3 ( µ21X−1L + λµ2−1Ψ2D1 ) D2 + q3X−2RD1 − q5µ2−1µ21X−2LD1 = q3µ21X−1LD2 + q2λµ2−1(X−2LX2R + qµ2−2Ψ3)D2D1 + q3X−2RD1 − q5µ2−1µ21X−2LD1 = q3µ21X−1LD2 + q3µ2−1 ( µ22 − 1 ) X−2LD1 + q3λµ2−1µ 2 −2Ψ3D2D1 + q3X−2RD1 − q5µ2−1µ21X−2LD1 (4.9′a) = q3µ21X−1LD2 − q3µ2−1X−2LD1 − q5µ2−1µ21X−2LD1 + q2[2]qX−2RD1,[ q−1X−1LD1,X1RD2 ] q2 = q−1X−1L [D1,X1R ]q4D2 = λ−1X−1L (( q2µ21 − 1 ) − q4 ( µ21 − 1 )) D2 = qX−1L ( q2 + 1− q2µ21 ) D2 = q ( q[2]q − q2µ21 ) X−1LD2 = q2[2]qX−1LD2 − q3µ21X−1LD2, (4.9′b) [X−1RD1,−qX−2LD−1]q2 = −qX−1R [D1,X−2LD−1]q − q 2[X−1R ,X−2LD−1]qD1 = −qX−1RX−2L [D1,D−1]q + q5µ2−1µ 2 1X−2LD1 = q5µ2−1µ 2 1X−2LD1, (4.9′c)[ q−1X−1LD1,−qX−2LD−1 ] q2 = −q[X−1L ,X−2LD−1]qD1 = −q2X−2L [X−1L ,D−1]D1 = q3X−2Lµ 2 −1D1 = q3µ2−1X−2LD1. (4.9′d) It is easy to see that [e1, e2]q2 = [e1,1, e−1,2]q2 = −q−2[2]−1q τ1τ −1 −1 [X−1RD1+ q−1X−1LD1,X1RD2− qX−2LD−1]q2 . So from (4.9′a)–(4.9′d), we obtain [e1, e2]q2 = −τ1τ−1−1 (X−1LD2 + X−2RD1) = e1,2. (2) To prove e−i,j = [e−i,j−1, ej ]q for 3 ≤ i + 2 ≤ j ≤ n, we compute the following four brackets first. By (2.4), (5.3) and Lemma 5.2, for 3 ≤ i+ 2 ≤ j ≤ n, we get [XiRDj−1,Xj−1RDj ]q = XiR [Dj−1,Xj−1RDj ]q = XiR [Dj−1,Xj−1R ]q2Dj = q2XiRDj , (4.10′a) [[Dj−1,Ψi+1]qD−i,Xj−1RDj ]q = [[Dj−1,Ψi+1]q,Xj−1RDj ]qD−i = [Dj−1, [Ψi+1,Xj−1RDj ]]q2D−i + [[Dj−1,Xj−1RDj ]q,Ψi+1]qD−i = [Dj−1, [Ψi+1,Xj−1RDj ]]q2D−i + [[Dj−1,Xj−1R ]q2Dj ,Ψi+1]qD−i = [Dj−1, [Ψi+1,Xj−1RDj ]]q2D−i + q2[Dj ,Ψi+1]qD−i, (4.10′b) [XiRDj−1, qX−jLD1−j ]q = q[XiRDj−1,X−jL ]D1−j = 0, (4.10′c) [[Dj−1,Ψi+1]qD−i, qX−jLD1−j ]q = [[Dj−1,Ψi+1]q, qX−jLD1−j ]qD−i = q[Dj−1, [Ψi+1,X−jLD1−j ]]q2D−i + q[[Dj−1,X−jLD1−j ]q,Ψi+1]qD−i = [Dj−1, [Ψi+1,Xj−1RDj ]]q2D−i + q[[Dj−1,X−jL ]D1−j ,Ψi+1]qD−i = [Dj−1, [Ψi+1,Xj−1RDj ]]q2D−i. (4.10′d) Realization of Uq(sp2n) within the Differential Algebra on Quantum Symplectic Space 19 From (4.10′a)–(4.10′d), it is easy to see that [e−i,j−1, ej ]q = [e−i,j−1, e1−j,j ]q = (−1)i+jq−4[XiRDj−1 − [Dj−1,Ψi+1]qD−i,Xj−1RDj − qX−jLD1−j ]q = (−1)i+jq−2(XiRDj − [Dj ,Ψi+1]qD−i) = e−i,j . (3) To prove ei,j = [ei,j−1, ej ]q for 3 ≤ i+ 2 ≤ j ≤ n, we compute the following four brackets first. By (2.2), (5.8), (5.9), Lemmas 5.2 and 5.3, for 3 ≤ i+ 2 ≤ j ≤ n, we get [X−iLDj−1,Xj−1RDj ]q = X−iL [Dj−1,Xj−1R ]q2Dj = q2X−iLDj , (4.11′a)[ qi−1X1−jR [Φi,X−iL ]q,Xj−1RDj ] q = qi−1[X1−jR ,Xj−1RDj ]q[Φi,X−iL ]q = qi−1(q2X−jR − q j+2Λ2 1−jµ 2 j−1X−jL)[Φi,X−iL ]q = qi+1X−jR [Φi,X−iL ]q − qi+j+1Λ2 1−jµ 2 j−1X−jL [Φi,X−iL ]q, (4.11′b) [X−iLDj−1,−qX−jLD1−j ]q = −qX−iLX−jL [Dj−1,D1−j ]q = 0, (4.11′c)[ qi−1X1−jR [Φi,X−iL ]q,−qX−jLD1−j ] q = −qi[X1−jR ,X−jLD1−j ]q[Φi,X−iL ]q = qi+j+1Λ2 1−jµ 2 j−1X−jL [Φi,X−iL ]q. (4.11′d) From (4.11′a)–(4.11′d), it is easy to see that [ei,j−1, ej ]q = (−1)j+1q−2τ1τ −1 −1 [X−iLDj−1+ qi−1X1−jR [Φi,X−iL ]q,Xj−1RDj− qX−jLD1−j ]q = (−1)j+1τ1τ −1 −1 ( X−iLDj + qi−1X−jR [Φi,X−iL ]q ) = ei,j . (4) To prove ej−1,j = [ej−1, ej−2,j ]q for 3 ≤ j ≤ n, we compute the following four brackets first. By (2.2) and Lemmas 5.2–5.4, for 3 ≤ j ≤ n, we get [Xj−2RDj−1,X2−jLDj ]q = q−1[Xj−2R ,X2−jLDj ]q2Dj−1 = q−1[Xj−2R ,X2−jL ]qDjDj−1 = 0, (4.12′a) qj−3[Xj−2RDj−1,X−jR [Φj−2,X2−jL ]q]q = qj−2X−jR [Xj−2R , [Φj−2,X2−jL ]q]Dj−1 = −qj−3X−jR [[Φj−2,Xj−2R ]q,X2−jL ]q2Dj−1 = −qj−1X−jR [ Λ2 j−2D2−j ,X2−jL ] q2 Dj−1 = −qj−1Λ2 j−2X−jR [D2−j ,X2−jL ]q2Dj−1 = −qjΛ2 j−2X−jRDj−1, (4.12′b) −q[X1−jLD2−j ,X2−jLDj ]q = −q[X1−jLD2−j ,X2−jL ]qDj = −qX1−jL [D2−j ,X2−jL ]q2Dj = −q2X1−jLDj , (4.12′c) −qj−2 [ X1−jLD2−j ,X−jR [Φj−2,X2−jL ]q ] q = −qj−2X−jR [ X1−jLD2−j , [Φj−2,X2−jL ]q ] q = −qj−2X−jRX1−jL [ D2−j , [Φj−2,X2−jL ]q ] q = qj−1X−jRX1−jL [ Φj−2, [X2−jL ,D2−j ]q−2 ] q2 = qj−1X−jRX1−jL [ Φj−2,−q−1 ] q2 = qj−1λX−jRX1−jLΦj−2. (4.12′d) From (4.12′a)–(4.12′d), it is easy to see that [ej−1, ej−2,j ]q = (−1)jq−2τ1τ −1 −1 [ Xj−2RDj−1 − qX1−jLD2−j ,X2−jLDj + qj−3X−jR [Φj−2,X2−jL ]q ] q = (−1)j+1τ1τ −1 −1 ( X1−jLDj + qj−2X−jR ( Λ2 j−2Dj−1 − λq−1X1−jLΦj−2 )) = (−1)j+1τ1τ −1 −1 ( X1−jLDj + qj−2X−jR [Φj−1,X1−jL ]q ) = ej−1,j . 20 J. Zhang and N. Hu (5) To prove ej,j = [2]−1q [e1,j , e−1,j ] for 2 ≤ j ≤ n, we compute the following four brackets first. By Lemma 5.2, (5.1), (5.5) and (5.11), for 2 ≤ j ≤ n, we get [X−1LDj ,X1RDj ] = q−1[X−1L ,X1R ]qD 2 j = −λX−1LX1RD 2 j , (4.13′a) −[X−1LDj , [Dj ,Ψ2]qD−1] = −q−1[X−1L , [Dj ,Ψ2]qD−1]qDj = −[Dj ,Ψ2]q[X−1L ,D−1]Dj = qµ2−1[Dj ,Ψ2]qDj = qµ2−1[Dj , q −1µ−2−1(Ψ1 − X−1LX1R)]qDj = [Dj ,Ψ1]qDj + λX−1LX1RD 2 j , (4.13′b) [X−jRD1,X1RDj ] = X−jR [D1,X1RDj ]q = X−jR [D1,X1R ]q2Dj = q2X−jRDj , (4.13′c) −[X−jRD1, [Dj ,Ψ2]qD−1] = −q[X−jR , [Dj ,Ψ2]qD−1]q−1D1 = −q[X−jR , [Dj ,Ψ2]q]D−1D1 = 0. (4.13′d) It is easy to see that [e1,j , e−1,j ] = (−1)i+jq−2 [ τ1τ −1 −1 (X−1LDj + X−jRD1),X1RDj − [Dj ,Ψ2]qD−1 ] = (−1)i+jq−2τ1τ −1 −1 [ X−1LDj + X−jRD1,X1RDj − [Dj ,Ψ2]qD−1 ] . So from (4.13′a)–(4.13′d), we get [e1,j , e−1,j ] = (−1)i+jτ1τ −1 −1 ( X−jRDj + q−2[Dj ,Ψ1]qDj ) = [2]qej,j . We complete the proof. � Hence, we can obtain the operators e±i,j from ei by the same inductive formulas that we used to get E±i,j from Ei. In other words, all the positive root vectors E±i,j of Uq(sp2n) can be realized by the operators e±i,j in the subalgebra U2n q of Diff(X ). Acknowledgements The authors would appreciate the referees for their useful comments and good suggestions for improving the paper. The first author is supported by the NSFC (Grants No. 11101258 and No. 11371238). The second author is supported by the NSFC (Grant No. 11771142). 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Realization of Uq(sp₂n) within the Differential Algebra on Quantum Symplectic Space

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