Enveloping Skewfields of the Nilpotent Positive Part and the Borel Subsuperalgebra of Osp(1,2N) Jacques Alev, François Dumas

Enveloping skewfields of the nilpotent positive part and the Borel subsuperalgebra of osp(1,2n) Jacques Alev, François Dumas

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Jacques Alev, François Dumas. Enveloping skewfields of the nilpotent positive part and the Borel subsuperalgebra of osp(1,2n). Contemporary mathematics, American Mathematical Society, 2019, Rings, Modules and Codes Fifth International Conference Noncommutative Rings and their Applica- tions June 12–15, 2017 University of Artois, Lens, France, 727, pp.7-23. 10.1090/conm/727/14621. hal-02401624

HAL Id: hal-02401624 https://hal.archives-ouvertes.fr/hal-02401624 Submitted on 10 Dec 2019

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ENVELOPING SKEWFIELDS OF THE NILPOTENT POSITIVE PART AND THE BOREL SUBSUPERALGEBRA OF osp(1, 2n)

JACQUES ALEV AND FRANC¸OIS DUMAS

Abstract. We study an analogue of the Gelfand-Kirillov property for some Lie superalgebras. More precisely, we consider in the classical simple orthosymplectic Lie superalgebra osp(1, 2n) of dimension n2 +3n the positive nilpotent subsuperalgebra n+ of dimension n2 + n and the solvable Borel subsuperalgebra b+ of dimension n2 + 2n. We prove that the enveloping algebras U(n+) and U(b+) are rationally equivalent to some super-analogues of Weyl algebras over polynomial superalgebras.

Introduction In the classical context of the seminal paper [7], an algebraic finite- dimensional complex Lie algebra g is said to satisfy the Gelfand-Kirillov property when its enveloping algebra U(g) is rationaly equivalent to a Weyl algebra An(K) over a commutative purely transcendental field extension K of C. It was proved that this property is satisfied when g is nilpotent ([7]), or more generally solvable ([5], [10], [12]). This profound problem gave rise to an abundant literature with many developments within Lie theory itself (see references in [3] or [16]) or for variants concerning quantum groups, invariant algebras or other classes of interesting noncommutatives algebras. The study of an analogue of the Gelfand-Kirillov property for Lie superalgebras was explored in [13] and [2]. In this context, it is relevant to introduce for any integer n ≥ 1 the noncommutative polynomial superal- n gebra Oe(C ) and an analogue Aen(C) of the Weyl algebra. By definition n Oe(C ) is the algebra generated over C by n indeterminates y1, . . . , yn with relations yiyj + yjyi = 0 for all 1 ≤ i 6= j ≤ n. Then Aen(C) is the algebra n generated over Oe(C ) by n other indeterminates x1, . . . , xn with relations xiyj + yjxi = xixj + xjxi = 0 for all 1 ≤ i 6= j ≤ n and xiyi − yixi = 1 for any 1 ≤ i ≤ n. These algebras are noetherian domains and then they admit a skewfield of fractions. Otherwise the only case in the classification of classical simple finite dimensional complex Lie superalgebras g where the

Date: April 23, 2018. 2010 Mathematics Subject Classification. Primary 17B35; Secondary 16S30, 16S85, 16K40. Key words and phrases. Simple Lie superalgebra, enveloping algebra, Gelfand-Kirillov hypothesis, Weyl algebra, skewfield. 1 2 JACQUES ALEV AND FRANÇOIS DUMAS enveloping algebra U(g) is a domain is when g is an orthosymplectic Lie superalgebra osp(1, 2n) (see [4], [9]). This is the case we consider here. For any integer n ≥ 1, the superalgebra g = osp(1, 2n) is of dimension 2 2n + 3n. In its Z2-graded decomposition g = g0 ⊕ g1, the even part g0 is isomorphic to the symplectic Lie algebra sp(2n) of dimension 2n2 + n, and the odd part g1 is a vector space of dimension 2n. There is a natural definition of the nilpotent positive part n+ = osp+(1, 2n), which is a nilpotent 2 subsuperalgebra of g of dimension n + n. In its Z2-graded decomposition n+ = n+ ⊕ n+, the even part n+ ' sp+(2n) is just the nilpotent positive 0 1 0 part of dimension n2 in the ordinary triangular decomposition of the Lie algebra g ' sp(2n), and the odd part n+ is a subspace of dimension n in g . 0 1 1 Denoting by h the Cartan Lie subalgebra of g0, the Borel subsuperalgebra b+ of g is defined as the subsuperalgebra generated by n+ and h. We have b+ = n+ ⊕h = b+ ⊕n+, where the even part b+ = n+ ⊕h is the positive Borel 0 1 0 0 Lie subalgebra of g ' sp(2n). In particular, b+ and b+ are of dimensions 0 0 n2 + n and n2 + 2n respectively. The enveloping algebras U(n+) and U(b+) can be described as iterated skew polynomial algebras; then they are noetherian domains and therefore we can consider their skewfields of fractions. We prove in the paper:

Main theorem. For any n ≥ 2, the enveloping algebras of the subsu- peralgebras n+ and b+ of osp(1, 2n) satisfy the following isomorphisms:

+ n (i) Frac U(n ) ' Frac An(n−1)/2(C)⊗Oe(C ) , and its center is a purely transcendental extension of dimension n over C. + (ii) Frac U(b ) ' Frac An(n−1)/2(C) ⊗ Aen(C) , and its center is C. The method we use to prove this theorem allows to recover as a corollary the classical Gelfand-Kirillov property for the even parts, with already known descriptions of Frac U(n+) as a Weyl skewfield D (K) over a 0 n(n−1)/2 commutative field K which is a purely transcendental extension of dimension n over , and of Frac U(b+) as a Weyl skewfield D ( ) over . C 0 n(n+1)/2 C C The particular case n = 2 of the main theorem was previously proved in Proposition 3.2 and Theorem 3.4 of [2]. The natural and probably difficult question of an extension of the main theorem to the enveloping algebra of the Lie superalgebra osp(1, 2n) itself must be appreciated by recalling that even for the even part the answer is unknown, the case of sp(2n) being one of the situations where the original Gelfand-Kirillov conjecture remains open (see [16]). The paper is in three parts. The first one introduces the analogues of Weyl algebras for polynomial superalgebras and some canonical form of their skewfields of fractions. The second part describes the enveloping algebras U(n+) and U(b+) as iterated skew polynomial algebras. The last section contains the proof of the main theorem, based on suitable rational changes of generators reducing step by step the commutation relations. ENVELOPING SKEWFIELDS 3

1. Analogues of Weyl algebras for polynomial superalgebras 1.1. Polynomial superalgebras.

n 1.1.1. Notations. It is usual in quantum group theory to denote by OΛ(C ) the algebra of polynomials in n indeterminates y1, . . . , yn with coefficients in C and noncommutative multiplication twisted by relations yiyj = λijyjyi for any 1 ≤ i, j ≤ n, where Λ = (λij) is a n×n multiplicatively skew-symmetric matrix with entries in C. In the particular cases where all λij = 1, or λij = −1 for all i 6= j, we use respectively the two following notations: n (i) O(C ) is the commutative algebra of polynomials in n indeterminates y1, y2, . . . , yn with coefficients in C. n (ii) Oe(C ) is the noncommutative algebra of polynomials in n indeterminates y1, y2, . . . , yn with coefficients in C and product twisted by relations yiyj = −yjyi for any 1 ≤ i 6= j ≤ n.

β β βn 1.1.2. Superalgebra structure. The monomials y1 1 y2 2 ··· yn whose total degree β1 + β2 + ··· + βn is even generate a commutative subalgebra R0 of n n Oe(C ). Denoting by R1 the R0-submodule of Oe(C ) generated by y1, . . . , yn, n the C-submodules R0 and R1 satisfy Oe(C ) = R0 ⊕ R1 and RiRj ⊆ Ri+j (with indices taken modulo 2). This gives rise to a structure of superalgebra n n on Oe(C ). The algebras Oe(C ) being noetherian domains, they admit skew ﬁelds of fractions. The following proposition gives an alternative form of n 2 Oe(C ) in terms of planes Oe(C ) up to rational equivalence. 1.1.3. Proposition. For any integer n ≥ 2, we have: n 2 ⊗p If n = 2p, then Frac (Oe(C )) ' Frac (Oe(C ) ) n 2 ⊗p If n = 2p + 1, then Frac (Oe(C )) ' Frac (Oe(C ) ⊗ O(C)) n Proof. Suppose that n ≥ 3 and deﬁne from generators y1, . . . , yn of Oe(C ) 0 0 with yiyj = −yjyi for all 1 ≤ i 6= j ≤ n the monomials: y1 = y1, y2 = y2 0 0 0 0 0 and yi = y1y2yi for any 3 ≤ i ≤ n. They satisfy y1y2 = −y2y1 and 0 0 0 0 0 0 0 0 0 0 0 0 y1yi = yiy1, y2yi = yiy2, yiyj = −yjyi for all 3 ≤ i 6= j ≤ n.

n 0 0 0 It is clear that the subﬁeld of Frac Oe(C ) generated by y1, y2, . . . , yn is equal n n 2 n−2 to Frac Oe(C ). We deduce that Frac Oe(C ) ' Frac (Oe(C ) ⊗ Oe(C )), and the result follows by iteration. 1.2. Weyl algebras for polynomial superalgebras.

1.2.1. Notations. The definition of quantum differential calculi on various quantum algebras with adapted de Rham complexes gives rise to suitable n versions of quantum Weyl algebras. In the case of quantum spaces OΛ(C ), q,Λ these algebras An (introduced in [1] and [19] and studied in many papers) take in consideration both the quantization parameters Λ = (λij) of the 4 JACQUES ALEV AND FRANÇOIS DUMAS space and the quantization parameters q = (q1, . . . , qn) of the “differential” operators on this space. We are interested here in two particular cases:

(i) For λij = qi = 1 for all 1 ≤ i, j ≤ n, we ﬁnd the usual Weyl algebra An(C), that is the algebra of polynomials in 2n indeterminates x1, . . . , xn, y1, . . . , yn with coeﬃcients in C and a noncommutative product twisted by relations: ( x x − x x = y y − y y = x y − y x = 0 (1 ≤ i 6= j ≤ n), i j j i i j j i i j j i (1) xiyi − yixi = 1 (1 ≤ i ≤ n).

We also introduce the subalgebra Un(C) of An(C) generated by y1, . . . , yn and w1, . . . , wn with wi = yixi for any 1 ≤ i ≤ n. We have: ( w w − w w = y y − y y = w y − y w = 0 (1 ≤ i 6= j ≤ n), i j j i i j j i i j j i (2) wiyi − yiwi = yi (1 ≤ i ≤ n).

(ii) For qi = 1 and λij = −1 for all 1 ≤ i 6= j ≤ n, we denote by Aen(C) the algebra of polynomials in 2n indeterminates x1, . . . , xn, y1, . . . , yn with coeﬃcients in C and a noncommutative product twisted by: ( x x + x x = y y + y y = x y + y x = 0 (1 ≤ i 6= j ≤ n), i j j i i j j i i j j i (3) xiyi − yixi = 1 (1 ≤ i ≤ n).

We denote by Uen(C) the subalgebra of Aen(C) generated by y1, . . . , yn and w1, . . . , wn with wi = yixi for any 1 ≤ i ≤ n. We have:  y y + y y = 0 (1 ≤ i 6= j ≤ n),  i j j i wiwj − wjwi = wiyj − yjwi = 0 (1 ≤ i 6= j ≤ n), (4)  wiyi − yiwi = yi (1 ≤ i ≤ n).

Λ 1.2.2. Remarks. The algebras Aen(C) are the algebras denoted by Sn,n and studied in [17] and [18] when all non diagonal entries of Λ are equal to −1. Section 11 of [19] gives a slightly diﬀerent presentation of these algebras as “generalized enveloping algebras”; with the notations of this article, case (i) corresponds to the values qij = pi = q = 1, and case (ii) is for qij = −1 and pi = q = 1. They also appear in example 2.1 of [8] (taking q = 1 and pij = −1). We refer to paragraph 1.3.3 of [17] for more ringtheoretical references and a survey, and only mention here that, for any integer n ≥ 1, the algebra Aen(C) is simple, has center C, and has the same Hochschild homology and cohomology as the usual Weyl algebra An(C). 1.2.3. Superalgebra structure. (i) Let g0 and g1 be two C-vector spaces of dimension n with respective basis {z1, . . . , zn} and {y1, . . . , yn}. Let k0 be a linear extension of g0 of dimension 2n with basis {z1, . . . , zn, w1, . . . , wn}. Then we deﬁne a Lie superalgebra structure g = g0 ⊕ g1 of dimension 2n and a Lie superalgebra structure k = k0 ⊕ g1 of dimension 3n by setting for the values of the super-bracket on these generators: ENVELOPING SKEWFIELDS 5 ( {yi, yi} = zi, [wi, yi] = yi, [wi, zi] = 2zi for any 1 ≤ i ≤ n, the brackets are zero for all other pairs of generators.

By construction g0 is a subsuperalgebra of k0 and g = g0 ⊕ g1 is a subsuperalgebra of k. It is clear by PBW theorem (see Theorem 6.1.2 in [14]) that the enveloping algebra U(k) is isomorphic to Uen(C) of GK-dim 2n. Moreover the subalgebra U(k0) is isomorphic to Un(C) of GK-dim 2n, and the subalgebra n U(g) is isomorphic to Oe(C ) of GK-dim n. (ii) We have for the usual Weyl algebra the classical isomorphism An(C) ' ⊗n ⊗n A1(C) . We also have Aen(C) ' Ae1(C)b in the tensor category of superalgebras assigning the parity 1 to the generators xi and yi. (iii) All isomorphims and tensor products considered in the following are in the tensor category of associative algebras. 1.2.4. Remark. All algebras introduced in 1.2.1 can be described as iterated skew polynomial algebras over C; then they are noetherian domains, and therefore they admit a skewfield of fractions. The skewfield Frac An(C) = Frac Un(C) is the well known Weyl skewfield Dn(C), with center C. Simi- larly we deduce from (4) that Frac Aen(C) = Frac Uen(C), and it is proved in Proposition 3.3.1 of [18] that its center is also C. In parallel with Proposition 1.1.3, we have for these skewfields the following canonical form. 1.2.5. Theorem. For any integer n ≥ 1, we have the following isomorphisms: ⊗p (i) If n = 2p is even, then Frac Aen(C) = Frac (Ae2(C) ). ⊗p (ii) If n = 2p + 1 is odd, then Frac Aen(C) = Frac (Ae2(C) ⊗ A1(C)).

Proof. We can suppose n ≥ 1. We consider the subalgebra Uen(C) defined in 1.2.1 with generators w1, . . . , wn, y1, . . . yn and relations (4). As in the proof 0 0 0 of Proposition 1.1.3, we introduce: y1 = y1, y2 = y2 and yi = y1y2yi for 3 ≤ 0 0 i ≤ n. Then we define: w1 = w1 −(w3 +···+wn), w2 = w2 −(w3 +···+wn) 0 and wi = wi for 3 ≤ i ≤ n. Obvious calculations give: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y1y2 = −y2y1, w1w2 = w2w1, y1w2 = w2y1, y2w1 = w1y2, 0 0 0 0 0 0 (5) [w1, y1] = y1, [w2, y2] = y2. 0 0 0 0 0 0 0 0 0 0 0 0 y1yj = yjy1, y2yj = yjy2, y1wj = wjy1, 0 0 0 0 0 0 0 0 0 0 0 0 for 3 ≤ j ≤ n. (6) y2wj = wjy2, w1wj = wjw1, w2wj = wjw2, 0 0 We calculate for ` = 1 or 2 and any 3 ≤ j ≤ n the commutator: [w`, yj] = Pn [w`, y1y2yj] − i=3[wi, y1y2yj] = y1y2yj − [wj, y1y2yj] = 0 to conclude : 0 0 0 0 0 0 0 0 yjw1 = w1yj and yjw2 = w2yj for any 3 ≤ j ≤ n. (7) Moreover: 0 0 0 [wj, yj] = yj for any 3 ≤ j ≤ n, (8) 0 0 0 0 0 0 0 0 0 0 0 0 yiyj = −yjyi, wiwj = wjwi, wiyj = yjwi, for 3 ≤ i 6= j ≤ n. (9) 0 0 0 0 We denote by V the subalgebra of Uen(C) generated by y1, . . . , yn,w1, . . . , wn, 0 0 0 0 0 by W the subalgebra of V generated by y1, , y2, w1, w2, and by W the 6 JACQUES ALEV AND FRANÇOIS DUMAS

0 0 0 0 subalgebra of V generated by y3, . . . , yn, w3, . . . , wn. It follows from (5) 0 00 that W ' Ue2(C), from (8) and (9) that W ' Uen−2(C), and from (7) 0 and (9) that V ' W ⊗ W . Since it is clear that Frac V = Frac Uen(C), we deduce that Frac Uen(C) ' Frac (Ue2(C) ⊗ Uen−2(C)). We conclude that Frac Aen(C) ' Frac (Ae2(C) ⊗ Aen−2(C)) and ﬁnish by induction on n.

2. The Lie superalgebra osp(1, 2n) and its enveloping algebra 2.1. Generators and relations. We refer for instance to [6], [15], [2]. The baseﬁeld is C. For any n ≥ 1, the odd part g1 of g = osp(1, 2n) is a vector space of dimension 2n and the even part g0 is isomorphic to the symplectic Lie algebra sp(2n) of dimension 2n2 + n. As a Lie superalgebra, osp(1, 2n) ± is generated by the elements bi (1 ≤ i ≤ n) of a basis of g1. Then the ± ± + − elements {bj , bk } (1 ≤ j ≤ k ≤ n) and {bj , bk } (1 ≤ j, k ≤ n) form a basis of g0. The brackets in osp(1, 2n) are given by the “para-Bose” relations on the generators: ξ η η ξ [{bj , bk}, b`] = ( − ξ)δj`bk + ( − η)δk`bj (10)

ξ η ϕ ξ ϕ η ϕ [{bi , bj }, {bk, b` }] = ( − η)δjk{bi , b` } + ( − ξ)δik{bj , b` } ξ η + (ϕ − η)δj`{bi , bk} + (ϕ − ξ)δi`{bj , bk}, (11) with 1 ≤ i, j, k, ` ≤ n and , ϕ, ξ, η are ±. By PBW theorem, the enveloping algebra U(osp(1, 2n)) is generated by the 2n2 + n elements: ± 1 − + bi and ki := 2 {bi , bi } for 1 ≤ i ≤ n, (12) ± 1 ± ± aij := 2 {bi , bj } for 1 ≤ i < j ≤ n, (13) 1 − + 1 + − sij := 2 {bi , bj } and tij := 2 {bi , bj } for 1 ≤ i < j ≤ n, (14) of osp(1, 2n). In particular, denoting ± 1 ± ± ci := 2 {bi , bi } for 1 ≤ i ≤ n, (15) the enveloping algebra U(sp(2n)) of the even part g0 is the subalgebra of 2 ± 2 ± U(osp(1, 2n)) generated by the 2n + n elements (bi ) = ci , ki for 1 ≤ ± i ≤ n, and aij, sij, tij for 1 ≤ i < j ≤ n. The commutation relations in the associative algebra U(osp(1, 2n)) are deduced from (10) and (11) taking {x, y} = xy + yx if x, y ∈ g1, and [x, y] = xy − yx otherwise. In particular commutation relations between the generators deﬁned by (12), (13), (14), (15) are obtained by injecting them in relations (10) and (11). For instance: + + + + + bi bj + bj bi = 2aij for all 1 ≤ i < j ≤ n, (16) + + + + [ci , bj ] = [ci , cj ] = 0 for all 1 ≤ i < j ≤ n, (17) + + + + [aij, b` ] = [aij, c` ] = 0 for all 1 ≤ i < j ≤ n, 1 ≤ ` ≤ n, (18) + + [aij, ak`] = 0 for all 1 ≤ i < j ≤ n, 1 ≤ k < ` ≤ n. (19) ENVELOPING SKEWFIELDS 7

Similarly with (10) and (14), we deduce for all 1 ≤ i < j ≤ n and 1 ≤ k ≤ n: + + + [tij, bj ] = bi and [tij, bk ] = 0 if k 6= j. (20)

The relations involving the tij’s are obtained from (11), (13) and (14), which give for 1 ≤ i < j ≤ n and 1 ≤ k < ` ≤ n : + 1 + + 1 + + [tij, ak`] = δjk 2 {bi , b` } + δj` 2 {bi , bk }, (21)

1 + − 1 + − [tij, tk`] = δjk 2 {bi , b` } − δi` 2 {bk , bj }, (22) or in other words: + [tij, ak`] = 0 if j 6= k, j 6= ` (1 ≤ i < j ≤ n, 1 ≤ k < ` ≤ n), (23) + + [tij, aj`] = ai` (1 ≤ i < j < ` ≤ n), (24) + + [tij, aij] = ci (1 ≤ i < j ≤ n), (25) + + [tij, akj] = aik if i < k (1 ≤ i < j ≤ n, 1 ≤ k < j ≤ n, ), (26) + + [tij, akj] = aki if k < i (1 ≤ i < j ≤ n, 1 ≤ k < j ≤ n, ). (27) [tij, tj`] = ti` (1 ≤ i < j < ` ≤ n), (28)

[tij, tki] = −tkj (1 ≤ k < i < j ≤ n), (29)

[tij, tk`] = 0 if j 6= k, i 6= ` (1 ≤ i < j ≤ n, 1 ≤ k < ` ≤ n). (30)

The action of the generators ki follows from (10) and (12):

[ki, kj] = 0 (1 ≤ i, j ≤ n), (31) + + ± [ki, bi ] = bi , [ki, bj ] = 0 (1 ≤ i 6= j ≤ n), (32) then with (13) and (14), for all 1 ≤ i < j ≤ n and 1 ≤ ` ≤ n:

 + +  [ki, aij] = aij, [ki, tij] = tij,  + +  [kj, aij] = aij, [kj, tij] = −tij, (33)  +  [k`, aij] = 0 if ` 6= i, j. [k`, tij] = 0 if ` 6= i, j.

2.2. Chevalley generators and Serre relations.√ Following [15] (up to a renormalization of the n-th generator by 2), we introduce: + ti := ti,i+1 for any 1 ≤ i ≤ n − 1, and tn := bn , (34) which satisfy:

[ti, tj] = 0 if |i − j| > 1 (1 ≤ i, j ≤ n), (35)

[ti, ti+1] = ti,i+2 (1 ≤ i ≤ n − 2), (36) + [tn−1, tn] = bn−1. (37) It also follows from (20) that for any 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ n: + + + [ti, bi+1] = bi and [ti, bj ] = 0 if j 6= i + 1. (38) 8 JACQUES ALEV AND FRANC¸OIS DUMAS

By straightforward calculations using (29), (35), (36), (37), the elements t1, . . . , tn deﬁned in (34) satisfy the following Serre identities which already appear as (2.11) in [15]: 2 2 ti ti+1 − 2titi+1ti + ti+1ti = 0 (1 ≤ i ≤ n − 1), (39) 2 2 ti ti−1 − 2titi−1ti + ti−1ti = 0 (2 ≤ i ≤ n − 1), (40) 3 2 2 3 tntn−1 − (tntn−1tn + tntn−1tn) + tn−1tn = 0. (41) 2.3. Enveloping algebra of the nilpotent subsuperalgebra n+.

2.3.1. Deﬁnition of n+. We deﬁne n+ := osp+(1, 2n) as the subsuperalgebra of osp(1, 2n) generated by Chevalley generators t1, t2, . . . , tn. By (34), + + + + + (37) and (38), n contains the elements b1 , b2 , . . . , bn . Hence by (13), n + contains the elements aij for 1 ≤ i < j ≤ n. It also follows inductively from + (28) and (36) that n contains the elements tij for 1 ≤ i < j ≤ n. Hence: n+ = n+ ⊕ n+, 0 1 where n+ is the -vector space with basis (b+) , and n+ ' sp+(2n) is 1 C i 1≤i≤n 0 + + the Lie subalgebra of g0 ' sp(2n) with basis (ci , aij, tij)1≤i

Then we add the generators tij for 1 ≤ i < j ≤ n inductively:

R1 := B[t1,2 ; d1,2][t1,3 ; d1,3] ··· [t1,n ; d1,n],

Ri := Ri−1[ti,i+1 ; di,i+1][ti,i+2 ; di,i+2] ··· [ti,n ; di,n], (2 ≤ i ≤ n − 2) + Rn−1 := Rn−2[tn−1,n ; dn−1,n] = U , ENVELOPING SKEWFIELDS 9 where the derivations dij are suitably deﬁned to account for relations (20) and (23) to (30). To sum up we conclude: + A = C[aij]1≤i

2.3.3. Restriction to the even part. Because of relations (17) and (18), we can consider in B the commutative subalgebra C generated over A by the + + 2 elements ck = (bk ) for 1 ≤ k ≤ n. It follows from previous results that: + + C = C[aij, ck ]1≤i

+ 2.4.1. Deﬁnition of b . The Cartan Lie subalgebra h of g0 is the Lie subalgebra with basis (ki)1≤i≤n, which is abelian by relation (31). The positive Borel subsuperalgebra b+ is deﬁned as the subsuperalgebra of osp(1, 2n) generated by n+ and h. We have clearly: b+ = n+ ⊕ h = b+ ⊕ n+, with b+ := n+ ⊕ h, 0 1 0 0 where the even part b+ is the positive Borel Lie subalgebra of g ' sp(2n). 0 0 2.4.2. Description of U(b+) as an iterated polynomial algebra. We denote + + + by Ub the enveloping algebra U(b ) of b . The commutation relations of + the ki’s with the generators of U given by formulas (31) to (33) allow an + easy description of Ub as the following iterated skew polynomial algebra: + + Ub = U [k1, ∆1][k2, ∆2] ··· [kn, ∆n], (49) + + where the derivations ∆i vanish on all kj’s and satisfy ∆i(bi ) = bi and + + + ∆i(bj ) = 0 for any 1 ≤ j 6= i ≤ n, ∆i(aij) = aij and ∆i(tij) = tij for + + 1 ≤ i < j ≤ n, ∆i(aji) = aji and ∆i(tji) = −tji for 1 ≤ j < i ≤ n, + ∆i(ak`) = 0 and ∆i(tk`) = 0 for k 6= i, ` 6= i. 2.4.3. Restriction to the even part. It follows from previous results that, with notations (49) and (48): U(b+) = U(n+)[k ; ∆ ] = C[t ; d ] [k ; ∆ ] . (50) 0 0 i i 1≤i≤n ij ij 1≤i

3. Proof of the main theorem 3.1. Some general technical results. We will use the following easy lem- mas. 10 JACQUES ALEV AND FRANC¸OIS DUMAS

3.1.1. Lemma. Let A be a commutative C-algebra, a an element of A, and R the iterated skew polynomial algebra generated over A by two generators x1 and x2 commuting with the elements of A and with relation x1x2 + x2x1 = 2a. Then the element y = x1x2 − a satisﬁes x1y = −yx1 and x2y = −yx2. Proof. Obvious calculation.

3.1.2. Lemma. Let A be a commutative C-algebra. Let (aij)1≤i

xjyi = xjx1xi − xja1i = (−x1xj + 2a1j)xi − a1ixj

= −x1xjxi + 2a1jxi − a1ixj = x1xixj − 2aijx1 + 2a1jxi − a1ixj

= (x1xi − a1i)xj − 2aijx1 + 2a1jxi = yixj − 2aijx1 + 2a1jxi. We use this identity and Lemma 3.1.1 to calculate:

yjyi = (x1xj − a1j)yi = x1(yixj − 2aijx1 + 2a1jxi) − a1jyi 2 = −yix1xj − 2aijx1 + 2a1jx1xi − a1jyi 2 = −yi(x1xj − a1j) − 2a1jyi − 2aijx1 + 2a1jx1xi 2 = −yiyj − 2a1jyi − 2aijx1 + 2a1jx1xi 2 = −yiyj + 2a1j(x1xi − yi) − 2aijx1. 2 We conclude that yjyi = −yiyj + 2a1ja1i − 2aijx1. 3.2. Enveloping skewﬁeld of the Lie subsuperalgebra n+. 3.2.1. Notations. Recalling the description (44), (45), (46) of U +, we introduce the skewﬁelds of rational functions: + K := Frac A = C(aij)1≤i

It is clear that we have with notation (42):

yk ∈ Bk, xk,i ∈ Bi, uk,i,j ∈ Bk−1 for all 1 ≤ k ≤ n, k ≤ i < j ≤ n. (56) 3.2.3. Lemma.

(i) The elements y1, y2, . . . , yn of B satisfy the relations:  y y = −y y for any 2 ≤ k ≤ n,  k−1 k k k−1 yky` = y`yk for all 1 ≤ k, ` ≤ n, ` 6= k ± 1,  + + ykaij = aijyk for all 1 ≤ k ≤ n, 1 ≤ i < j ≤ n.

(ii) For any 1 ≤ k ≤ n, the elements uk,i,j with k ≤ i < j ≤ n are in the 2 2 2 commutative subalgebra A[y1, y2, . . . , yk−1] of B, Proof. By iterative application of Lemma 3.1.2, we have for any 1 ≤ k ≤ n:

xk−1,k−1xk,i = −xk,ixk−1,k−1 for any 2 ≤ k ≤ i ≤ n, (57)

xk,ixk,j + xk,jxk,i = 2uk,i,j for all k ≤ i < j ≤ n. (58) We deduce by induction on ` that, for any 1 ≤ ` ≤ n:  y x = −x y for any ` + 1 ≤ i ≤ n,  ` `+1,i `+1,i ` y` xk,i = xk,i y` for any ` + 2 ≤ k ≤ i ≤ n, (59)  y` uk,i,j = uk,i,j y` for any 1 ≤ k ≤ i < j ≤ n. Point (i) of the lemma follows obviously from these relations and point (ii) follows by iterative definition (55) of the uk,i,j’s. 3.2.4. Proposition. (i) The subalgebra of B generated over C by y1, . . . , yn is isomophic to n OΛ(C ), where all entries of the symmetric matrix Λ = (λij)1≤i,j≤n are equal to 1 except λi,i+1 = −1 = λi+1,i for any 1 ≤ i ≤ n − 1. (ii) Its skewfield of fractions Q satisfies: + L = Q(aij)1≤i

0 0 0 0 0 0 0 0 0 0 0 0 y1y2 = −y2y1, y1yi = yiy1, y2yi = y2y1, for any 3 ≤ i ≤ n, 0 0 0 0 0 0 0 0 y3y4 = −y4y3, y3yi = yiy3, for any 5 ≤ i ≤ n, 2 We deduce that Q ' Frac (Oe(C ) ⊗ Tn−2) where Tn−2 is the subalgebra of 0 Q generated by the elements (yi)3≤i≤n. Applying the same reduction on the generators of Tn−2, we conclude by iteration and using Proposition 1.1.3 n that Q ' Frac (Oe(C )). 3.2.5. Lemma. For any 1 ≤ i ≤ n − 1 and any 1 ≤ m ≤ n, we have: 2 [ti, um,i,i+1] = xm,i (63) [ti, um,k,i+1] = um,k,i for m ≤ k < i ≤ n, (64)

[ti, um,i+1,`] = um,i,` for m < i + 1 < ` ≤ n, (65)

[ti, um,i+1,`] = 0 for m = i + 1 < ` ≤ n, (66)

[ti, um,k,`] = 0 for k 6= i + 1 and ` 6= i + 1, (67)

[ti, xm,i+1] = xm,i for i + 1 > m, (68)

[ti, xm,i+1] = 0 for i + 1 = m, (69)

[ti, xm,k] = 0 for k 6= i + 1. (70)

Proof. We study the action on B of the Chevalley generators t1, t2, . . . , tn−1 defined in 2.2. Let us recall that, by (38) we have: + + + [ti, bi+1] = bi and [ti, bk ] = 0 if k 6= i + 1. (71) Similarly by (23) to (27), we obtain: + + 2 [ti, ai,i+1] = (bi ) , (72) + + [ti, ak,i+1] = ak,i for 1 ≤ k < i ≤ n, (73) + + [ti, ai+1,`] = ai,` for i + 1 < ` ≤ n, (74) + [ti, ak,`] = 0 for k 6= i + 1 and ` 6= i + 1. (75) Then for fixed i, we prove the assertions of the lemma by induction on m. By (54), assertion (71) is the case m = 1 of assertions (68) to (70), and (72) to (75) are the case m = 1 of assertions (63) to (67). Suppose that the eight assertions are satisfied to some rank m ≥ 1. By (55), we have:

[ti, um+1,k,`] = [ti, um,m,k]um,m,` + um,m,k[ti, um,m,`] 2 2 − [ti, um,k,`]xm,m − um,k,`[ti, xm,m]. It follows from (69) and (70) that the fourth term is zero. Then using induction assumptions (63) to (67), tedious but straightforward calculations prove that these ﬁve assertions remain valid at the rank m + 1. Similarly by (55) we have:

[ti, xm+1,k] = [ti, xm,m]xm,k + xm,m[ti, xm,k] − [ti, um,m,k]. It follows from (69) and (70) that the ﬁrst term is zero. Then using induction assumptions (63) to (70), it is easy to check that: [ti, xm+1,i+1] equals 0 if ENVELOPING SKEWFIELDS 13 m + 1 = i + 1 and equals xm+1,i if i + 1 > m + 1, and [ti, xm+1,k] = 0 for any k 6= i + 1. This ﬁnishes the proof.

3.2.6. Corollary. Each generator tij for 1 ≤ i < j ≤ n commutes with each generator ym for 1 ≤ m ≤ n.

Proof. By (69) and (70), the Chevalley generators t1, t2, . . . , tn−1 commute with ym = xm,m for any 1 ≤ m ≤ n. By (28) and (36), each generator tij for 1 ≤ i < j ≤ n is an iterated bracket of t1, t2, . . . , tn−1. 3.2.7. Proposition. The skewﬁeld of fractions of U + satisﬁes: + F = Q (aij)1≤i

3.2.8. Lemma. Denote by (qi,j)1≤i

Then the qi,j’s commute with each other and with all elements of Q, and the subﬁeld of L generated over Q by the (qi,j)1≤i

[tij, qi,j] = 1, (78)

[tij, qk,`] = 0 if j 6= `, (79)

[tij, qk,j] = 0 if i < k, (80)

[tij, qk,j] = qk,i if i > k. (81) Proof. We proceed by induction on j − i. For j = i + 1, the identities directly follows from Lemma 3.2.5. For the general case, we deduced from (28) and (34) that ti,j+1 = [ti,j, tj,j+1] = [ti,j, tj]. Then we have [ti,j+1, qk,`] = [[ti,j, tj], qk,`] = [ti,j, [tj, qk,`]] − [tj, [ti,j, qk,`]] for any 1 ≤ k < ` ≤ n, and we 14 JACQUES ALEV AND FRANC¸OIS DUMAS

ﬁnish the proof by induction considering successively the diﬀerent cases for the indices (i, j) and (k, `).

3.2.10. Corollary. The elements (qi,j)1≤i

Proof. Follows obviously from Lemma 3.2.9 by induction on i.

3.2.11. Proposition. We have: F = Q(qij)1≤i

Proof. It’s clear by Lemma 3.2.8 that L = Q(qij)1≤i

Proof. By Corollary 3.2.10, the generators pij and qij commute with all elements of Q and the subalgebra W of F they generate is isomorphic to the usual Weyl algebra An(n−1)/2. Then the theorem follows from Proposition 3.2.11 and assertion (61) of Proposition 3.2.4. 3.2.13. Conclusion. We have proved the ﬁrst assertion of the main theorem.

3.3. Enveloping skewfield of the Lie subalgebra n+. 0 3.3.1. Notations. In order to determine the skewfield of fractions of the enveloping algebra of the even part n+ of n+ described in (48), we complete 0 the notations (51), (52), (53) and (61) by defining: 2 2 2 Q0 := C(y1, y2, . . . , yn) ⊂ Q, (83) + 2 + 2 + 2 L0 := K((b1 ) , (b2 ) ,..., (bn ) ) ⊂ L, (84) F := Frac U(n+) = L (t ; d ) ⊂ F. (85) 0 0 0 ij ij 1≤i

3.3.3. Proposition. F0 = Q0 (qij)1≤i

Proof. It is clear by previous lemma that L0 = Q0(aij)1≤i

3.3.4. Conclusion. The previous proposition showed that: + n Frac U(n0 ) ' Frac (An(n−1)/2(C) ⊗ O(C )). Hence we retrieve in the context of the classical Gelfand-Kirillov property the usual description of the enveloping skewﬁeld of the nilpotent Lie al- + gebra sp (2n) as a Weyl skewﬁeld Dn(n−1)/2 over a purely transcendental extension of dimension n over C.

3.4. Enveloping skewﬁeld of the Lie subsuperalgebra b+. Resuming all notations of the previous section and using (49) and (53), we set:

+ H := Frac Ub = F (k1, ∆1)(k2, ∆2) ··· (kn, ∆n). (86)

Our ﬁrst step is to determine the action of the generators k1, . . . , kn of the Cartan subalgebra h on the generators yi, pij, qij (1 ≤ i < j ≤ n) introduced in the description of F in Proposition 3.2.11.

3.4.1. Lemma. For any 1 ≤ ` ≤ n, we have the following identities: ( y` if m = `, [k`, ym] = m−`−1 and [k`, ym] = 0 if m < `. (87) 2 ym if m > `,

 [k`, q`,j] = −q`,j [k`, p`,j] = p`,j for any j > `, [k`, qi,`] = qi,` [k`, pi,`] = −pi,` for any i < `, (88) [k`, qi,j] = 0 [k`, pi,j] = 0 for any i 6= ` 6= j.

+ + Proof. We start with the action of k` on bi and ai,j for any 1 ≤ i < j ≤ n given by relations (32) and (33). Using the iterative deﬁnitions (54) and (55), we obtain for 1 ≤ m < ` the identities:  ( um,`,j if i = `, xm,` if i = `,  [k`, xm,i] = and [k`, um,i,j] = u if j = `, 0 if i 6= `, m,i,` 0 if i 6= ` 6= j. Similarly for m = `: ( ( x`,` if i = `, u`,`,j if i = `, [k`, x`,i] = and [k`, u`,i,j] = 0 if i 6= `, 0 if i 6= ` 6= j. 16 JACQUES ALEV AND FRANC¸OIS DUMAS

m−`−1 m−` Finally for m > `:[k`, xm,i] = 2 xm,i and [k`, um,i,j] = 2 um,i,j. Assertion (87) follows trivially from these relations since ym = xm,m. More- over, for 1 ≤ i, j ≤ n, we have by deﬁnition (77) and (82): −2 −3 −2 [k`, qi,j] = [k`, yi ui,i,j] = −2yi [k`, yi]ui,i,j + yi [k`, ui,i,j], (89) Pi−1 Pi−1 [k`, pi,j] = [k`, ti,j] − m=1[k`, qm,i]pm,j − m=1 qm,i[k`, pm,j]. (90) We deduce (88) applying case by case the previous identities. 3.4.2. Notation. An obvious observation arising from (88) is:

[k`, pi,jqi,j] = 0 for all 1 ≤ ` ≤ n, 1 ≤ i < j ≤ n. (91) These relations lead us to introduce the following elements of H: Pn P`−1 h` := k` + m=`+1 q`,mp`,m − m=1 qm,`pm,` for any 1 ≤ ` ≤ n, (92) which satisfy the relations described in the following lemma.

3.4.3. Lemma. For all 1 ≤ `, m ≤ n and 1 ≤ i < j ≤ n, we have:

[h`, qi,j] = [k`, pi,j] = [h`, hm] = 0 and [h`, ym] = [k`, ym]. Proof. By (92) we have: Pn P`−1 [h`, qi,j] = [k`, qi,j] + m=`+1 q`,m[p`,m, qi,j] − m=1 qm,`[pm,`, qi,j]. If i 6= ` and j 6= `, all brackets appearing in the right hand are zero by Corollary 3.2.10 and relation (88). If i = `, the equality becomes [h`, q`,j] = [k`, q`,j] + q`,j[p`,j, q`,j] = 0. If j = `, we obtain [h`, qi,`] = [k`, qi,`] − qi,`[pi,`, qi,`] = 0. The calculations are similar for pi,j. It is trivial 0 0 that [(pi,jqi,j), (pi0,j0 qi0,j0 )] = 0 for any 1 ≤ i < j ≤ n and 1 ≤ i < j ≤ n. Then [h`, hm] = 0 follows from the commutativity of the Cartan subalgebra and observation (91). By Proposition 3.2.11, the generators y1, . . . , yn commute with all pi,j’s and qi,j’s; then the last identity of the lemma is clear.

3.4.4. Proposition. Let W be the subalgebra of F generated over C by the n(n − 1) elements pij and qij. Let P be the subalgebra of H generated over C by y1, . . . , yn and h1, . . . , hn. Then H ' Frac (W ⊗ P ). Proof. It is clear by (86), (92) and Proposition 3.2.11 that the subﬁeld of H generated over F by h1, . . . , hn is equal to H. Then the result follows from Lemma 3.4.3 and Proposition 3.2.11.

0 0 3.4.5. Proposition. Let us deﬁne y1, . . . , yn in Q by: 0 0 −1 −1 −1 y1 = y1 and ym = y1 y2 ··· ym−1 ym for any 2 ≤ m ≤ n. (93) 0 0 0 Let P be the subalgebra of H generated over C by y1, . . . , yn and h1, . . . , hn. 0 0 Then P ' Uen(C) and H ' Frac (W ⊗ P ). ENVELOPING SKEWFIELDS 17

Proof. It is clear by (93) that Frac P 0 ' Frac P and H ' Frac (W ⊗ P 0). By tedious but simple calculations that we leave to the reader, we have: 0 0 0 0 0 0 0 y`ym = −ymy`, [h`, ym] = 0 and [h`, y`] = y` (94) for all 1 ≤ ` 6= m ≤ n. We conclude with point (ii) of 1.2.1.

3.4.6. Corollary. We have H ' Frac (An(n−1)/2(C)⊗Aen(C)), with center C.

Proof. We have already observed in Corollary 3.2.12 that W ' An(n−1)/2(C) and in Remark 1.2.4 that Frac Aen(C) ' Frac Uen(C). Then the isomorphism H ' Frac (An(n−1)/2(C) ⊗ Aen(C)) follows from previous proposition. Using the usual notation Dm(E) for the skewfield of fractions of a Weyl algebra Am(E) over a skewfield E, we have H ' Dn(n−1)(E) for E = Frac Aen(C). Its center satisfies Z(H) 'Z(Dn(n−1)(E)) = Z(E), and Z(E) = C by Proposition 3.3.1 of [18]. 3.4.7. Conclusion. The second assertion of the main theorem is proved. 3.5. Enveloping skewfield of the Lie subalgebra b+. 0 3.5.1. Notations. In order to determine the skewfield of fractions of the enveloping algebra of the even part b+ of b+ described in (50), we denote: 0 H := Frac U(b+) = F (k ; ∆ ) ··· (k ; ∆ ), (95) 0 0 0 1 1 n n where F0 is described by Proposition 3.3.3. 0 0 3.5.2. Proposition. We have H0 ' Frac (W ⊗P0), where P0 is the subalgebra 0 0 2 0 2 of P generated by (y1) ,..., (yn) , h1, . . . , hn, which is isomorphic to the enveloping algebra Un(C) defined in point (i) of 1.2.1.

Proof. By Proposition 3.3.3 and (92), the subﬁeld of H0 generated over F0 0 by h1, . . . , hn is equal to H0. Since Frac P0 = Q0 by (93), the result follows from Proposition 3.3.3.

3.5.3. Conclusion. Because of the isomorphism Frac Un(C) ' Frac An(C) observed in Remark 1.2.4 and the canonical isomorphism An(n+1)/2(C) ' An(n−1)/2(C) ⊗ An(C), the previous proposition proves that: Frac U(b+) ' Frac A ( ). 0 n(n+1)/2 C Then we retrieve in the context of classical Gelfand Kirillov property the usual description of the enveloping skewfield of the solvable Borel subalgebra of the Lie algebra sp(2n) as the Weyl skewfield Dn(n+1)/2(C). 3.6. Final observation. A thorough reading of the previous proofs shows that the isomorphisms we obtain at the level of the skewfields of fractions are already realized at the level of the localization of the algebras under consideration by the multiplicative set generated by the n elements y1, . . . , yn. Since these elements are homogeneous, this result is consistent with the notion of skewfield of fractions in the category of superalgebras. 18 JACQUES ALEV AND FRANÇOIS DUMAS

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(J. Alev) Universite´ de Reims, Laboratoire de Mathematiques´ (FRE 2011 - CNRS), Moulin de la Housse, B.P. 1039, 51687 Reims cedex 2 (France) E-mail address: [email protected]

(F. Dumas) Universite´ Clermont Auvergne, Laboratoire de Mathematiques´ Blaise Pascal (UMR 6620 - CNRS), 3 place Vasarely, CS 60026, 63178 Aubiere` cedex (France) E-mail address: [email protected]