Realization of $U_q({\mathfrak{sp}}_{2n})$ within the Differential Algebra on Quantum Symplectic Space

We realize the Hopf algebra $U_q({\mathfrak {sp}}_{2n})$ as an algebra of quantum differential operators on the quantum symplectic space $\mathcal{X}(f_s;\mathrm{R})$ and prove that $\mathcal{X}(f_s;\mathrm{R})$ is a $U_q({\mathfrak{sp}}_{2n})$-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 $U_q({\mathfrak {sp}}_{2n})$.


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) geometry, 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 C n q was built by Wess-Zumino [23] so as to extend the covariant coaction of the quantum group GL q (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 := X r (f s ; R) (adopting the notation in the book [10]). For the definition of polynomials f s in types ABCD, we refer to [10,Definitions 4,8,12 in Sections 9.2 and 9.3]. Roughly speaking, viewing U q (g) as the Hopf dual object of quantum group G q in types ABCD, one sees that the aforementioned quantum space X is a left U q (g)-module algebra. As a benefit of the viewpoint, this allows one to do the crossed product construction to enlarge the quantum enveloping algebra U q (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 U q (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 SO q (N ) (N = 2n + 1 or 2n) to define some quantum differential operators on the quantum Euclidean space R N q . Then he realized U q −1 (so N ) within the differential algebra Diff(R N q ) such that R N q is a left U q −1 (so N )-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 GL q (n) and SO q (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] introduced the notion of quantum divided power algebra A q (n) for q both generic and root of unity. He defined q-derivatives over A q (n) and realized the U -module algebra structure of A q (n) for U = U q (sl n ), u q (sl n ). A coherence realization of all the positive root vectors in terms of the quantum differential operators was provided (in the modified q-Weyl algebra W q (2n)) which are compatible with the actions of Lusztig's braid automorphisms [13]. Especially, this discussion 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 u q (sl n ).
For type C, it seems lack of corresponding discussions over the quantum symplectic space in the literature. Here we consider the quantum enveloping algebra U q (sp 2n ) with n ≥ 2 and its corresponding quantum symplectic space X (f s ; R). We assume that q is not a root of unity. We define the q-analogues ∂ i := ∂ q /∂x i of the classical partial derivatives and introduce left-and right-multiplication operators x i L and x i R as in [9]. Our discussion also does not use the R-matrix as a tool as in [2]. We consider the subalgebra U 2n q generated by some quantum differential operators in the quantum differential algebra Diff(X (f s ; 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 U 2n q and show X (f s ; R) is a U q (sp 2n )-module algebra. At last, we show that the positive root vectors of U q (sp 2n ) 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 (f s ; R) and derives some useful formulas. In Section 3, we define the quantum differential operators on X (f s ; R) and a subalgebra U 2n q of Diff(X (f s ; R)). We prove that the generators of U 2n q satisfy the Serre relations which implies that U 2n q is a quotient algebra of U q (sp 2n ). We show that X (f s ; R) is a U q (sp 2n )-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 U q (sp 2n ) from simple root vectors. In Section 5, we give a coherence realization for all the positive root vectors of U q (sp 2n ).
For simplicity, we write X for X (f s ; R). Let N 0 (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}.

Preliminaries
The following three lemmas can be checked directly and will be used many times in Sections 4 and 5.

2.2.
Recall that the simple roots of sp 2n are α 1 = 2 1 and α i = i − i−1 for 2 ≤ i ≤ n, where i = (δ 1i , . . . , δ ni ) and 1 , . . . , n form a canonical basis of R n . Note that here α 1 is chosen to be longer than other simple roots. Let ∆ + be the set of positive roots of sp 2n , then

2.3.
Recall that the quantum universal enveloping algebra U q (sp 2n ) generated by {E i , F i , K i , K −1 i , i ∈ I + } has the defining relations as follows: , and the Cartan matrix (a ij ) of sp 2n in our indices is Note that q 1 = q 2 , q i = q for 1 < i ≤ n. The relations (2.8) and (2.9) are usually called the Serre relations.
The algebra U q (sp 2n ) is a Hopf algebra equipped with coproduct ∆, counit ε and antipode S defined by 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 x i , i ∈ I, and defining relations: where Ω i := i≤j≤n q j−i x −j x j for i ∈ I + , and X is a vector space with basis x a −n −n · · · x an n | a −n , . . . , a n ∈ N 0 .
By definition, for 1 ≤ i ≤ n − 1, we have From relations (2.13) and (2.14), we can obtain the following identities: 16) and Set x a := x a −n −n · · · x an n and a := (a −n , a 1−n , . . . , a n ), where a −n , . . . , a n ∈ N 0 . We call the monomial x a i 1 i 1 · · · x a im im whose subscripts are placed in an increasing order a normal monomial. Write ε i = (0, . . . , 1, . . . , 0) ∈ R 2n with 1 in the i-position and 0 elsewhere. Then a = i∈I a i ε i . Set |a| = i∈I a i . Thus X = m X m is an N 0 -graded algebra with X m = Span k {x a | |a| = m}.
By induction and using relations (2.13)-(2.16), we get for i ∈ I + and m ∈ N 0 . Hence, for i ∈ I + , we have The following lemma will be used later.
Proof . We prove this lemma by induction on i from n to 1. From (2.13) and (2.16), we have The induction hypothesis completes the proof.
3 Quantum dif ferential operators on X (f s ; R) 3.1. We define some quantum analogs of differential operators on X .
Definition 3.1. For any normal monomial x a and i ∈ I, set Let Diff(X ) be the unital algebra of quantum differential operators on X generated by with i ∈ I. This algebra can be described precisely as the smash product of a quantum group D q and the symplectic space X , where the associative algebra D q generated by ∂ i 's (i ∈ I) as well as µ i '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 µ k µ l = µ l µ k , we write Applying the operators defined in Definition 3.2 to any normal monomial x a , and using Definition 3.1 and (2.17)-(2.19), we get The following two lemmas will be used later.
Lemma 3.4. For any two normal monomials x a , x b and i ∈ I + we have Proof . We prove this lemma by induction on |a|. For |a| = 1, write x a = x j , j ∈ I. The assertion 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 x a with |a| = m. Let x c be a normal monomial with |c| = m + 1. We can write x c = x j x a , where |a| = m and j is the smallest index in (c −n , . . . , c n ) such that c j = 0. Since x a x b can be written as a linear combination of normal monomials, by the induction hypothesis, we get Then Other relations can be proved similarly.
The following lemma can be easily checked by definition.
Now we state one of our main theorems.
Using the above two formulas and the identity which is easy to check, we can verify [e 1 , e 1,2 ] q −2 .x a = 0 by direct computation. So the relation (3.10) holds. Consider the first Serre relation (2.8) for i = 2, j = 1. We need to prove In order to verify the first Serre relation (2.8) for j = i ± 1 and i, j > 1, i.e., It is not hard to check (3.11) and (3.12) after applying their left hand sides to x a . For |i − j| > 2, the first Serre relation is [e i , e j ] = 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 U q (sp 2n ) as certain q-differential operators on X . In other words, X is a left U q (sp 2n )-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.(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 U q (sp 2n )-module algebra.

3.2.
We consider the decomposition of X into a direct sum of irreducible U q (sp 2n )-submodules. Recall that X = m∈N 0 X m , where X m is the subspace of homogeneous elements of degree m.
Proposition 3.8. The vector space X m is a finite-dimensional irreducible U q (sp 2n )-module with highest weight vector x m −n and highest weight mε n . Proof . The assertion follows at once from the facts that the symmetric powers of the vector representation of sp 2n are irreducible and the theory of finite-dimensional representations of U q (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.
4 Positive root vectors of U q (sp 2n ) We are going to list all positive root vectors of U q (sp 2n ) in U 2n q . We first recall some notions. Let m ij be equal to 2, 3, 4 when a ij a ji is equal to 0, 1, 2, respectively, where a ij are the entries of the Cartan matrix of sp 2n . The braid group B associated with sp 2n is the group generated by elements s 1 , . . . , s n subject to the relations where there are m ij s's on each side. Lusztig introduced actions of braid groups on U q (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 T i of U q (sp 2n ) which acts on the generators K j , E j , F j as The mapping s i → T i determines a homomorphism of the braid group B into the group of algebra automorphisms of U q (sp 2n ).  1. For i, j ∈ I + with |i − j| > 1, we have The Weyl group W of sp 2n generated by reflections w 1 , . . . , w n (corresponding to the simple roots of sp 2n ) has the longest element w 0 whose reduced expression is where γ i = w i w i−1 · · · w 1 · · · w i−1 w i (cf. [1]). Write w 0 = w i 1 w i 2 · · · w i N for this reduced expression. Then exhaust all positive roots of sp 2n .
are called positive root vectors of U q (sp 2n ) corresponding to the roots β r 's.
Set α i,i = 2 i and α ±l,k = ± l + k 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 w 0 as follows . . . , α n−1,n , α n−2,n , . . . , α 1,n , α n,n , α −1,n , α −2,n , . . . , α 1−n,n . Write It is clear that E 1,1 = E 1 and we will check in Corollary 4.5 that E α i = E i for all 1 < i ≤ n. Set By Definition 4.3, all the positive root vectors of U q (sp 2n ) associated to the above ordering of ∆ + are as follows, for any 1 < j ≤ n, The relation (4.6) is clear, since for i > 1 we have 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 To prove (4.8), we use induction on j − i. For j − i = 0, we have Suppose that (4.8) holds for some j − i − 1 > 0. Then by induction, we get So (4.8) holds.
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.
5 Realization of positive root vectors of U q (sp 2n ) In order to realize all the positive root vectors of U q (sp 2n ) directly and concisely as certain operators in Diff(X ), we introduce some new operators.
Definition 5.1. For i ∈ I + , set and Φ 0 := 0, Ψ n+1 := 0, Then we get The commutation relations in the following three lemmas will be used frequently in this section.
1. For k, l ∈ I and i ∈ I + , we have 3. For i, j ∈ I + with i = j, we have 4. For i ∈ I + , we have Then Proof . Applying both sides of each identity to any normal monomial x a , 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 Φ i and Ψ i satisfy the following commutation relations.
1. For i ∈ I + and t, k, l ∈ I with |t| < i and |k| > i, we have 3. For i, j ∈ I + with i ≤ j, we have 5. For i, k ∈ I + , we have From now on, Lemma 2.1 is frequently used without extra explanation.
Lemma 5.4. The operators X −i R for i ∈ I + satisfy the following commutation relations.
1. For i ∈ I + , k, l ∈ I with |k| < i and |l| = i, we have 3. For i, j ∈ I + with i < j, we have that is, (5.8) holds.
We are now in the position to realize all the positive root vectors E ±i,j of U q (sp 2n ) as e ±i,j in Diff(X ).
The next lemma says that the operators which realize the simple root vectors of U q (sp 2n ) defined in Definition 5.6 coincide with those defined in Definition 3.2.
Proof . From (2.20), it is easy to show that for any normal monomial x a . Then it yields from (2.19) that Write e i in terms of the new operators defined in Definition 5.1. By (5.4), we get and for i > 1 This completes the proof.
We complete the proof.
Hence, we can obtain the operators e ±i,j from e i by the same inductive formulas that we used to get E ±i,j from E i . In other words, all the positive root vectors E ±i,j of U q (sp 2n ) can be realized by the operators e ±i,j in the subalgebra U 2n q of Diff(X ).