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Enveloping skewfields of the nilpotent positive part and the Borel subsuperalgebra of osp(1,2n)
Jacques Alev, François Dumas
To cite this version:
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�
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 superalgebraosp(1,2n) of dimensionn2+ 3n the positive nilpotent subsuperalgebran+ of dimensionn2+nand the solvable Borel subsuperalgebrab+of dimensionn2+ 2n. We prove that the enveloping algebrasU(n+) andU(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 algebraU(g) is rationaly equivalent to a Weyl algebraAn(K) over a commutative purely transcendental field extensionK of C. It was proved that this property is satisfied wheng 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 su- peralgebras was explored in [13] and [2]. In this context, it is relevant to introduce for any integer n ≥ 1 the noncommutative polynomial superal- gebra O(e Cn) and an analogue Aen(C) of the Weyl algebra. By definition O(e Cn) is the algebra generated over C by nindeterminatesy1, . . . , yn with relations yiyj +yjyi = 0 for all 1 ≤i6=j ≤n. Then Aen(C) is the algebra generated over O(e Cn) 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 superalgebrasgwhere 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
enveloping algebra U(g) is a domain is when g is an orthosymplectic Lie superalgebraosp(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 2n2+ 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 defi- nition of the nilpotent positive partn+ =osp+(1,2n), which is a nilpotent subsuperalgebra of g of dimension n2+n. In its Z2-graded decomposition n+ = n+
0 ⊕n+
1, the even part n+
0 ' sp+(2n) is just the nilpotent positive part of dimensionn2 in the ordinary triangular decomposition of the Lie al- gebra g0'sp(2n), and the odd part n+
1 is a subspace of dimensionning1. Denoting by hthe 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+
0 ⊕n+
1, where the even partb+
0 =n+
0 ⊕his the positive Borel Lie subalgebra of g0 'sp(2n). In particular, b+
0 and b+ are of dimensions n2+nand n2+ 2nrespectively. The enveloping algebras U(n+) andU(b+) can be described as iterated skew polynomial algebras; then they are noe- therian 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- peralgebrasn+ and b+ of osp(1,2n) satisfy the following isomorphisms:
(i) FracU(n+)'Frac
An(n−1)/2(C)⊗O(e Cn)
, and its center is a purely transcendental extension of dimensionnoverC.
(ii) FracU(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 corol- lary the classical Gelfand-Kirillov property for the even parts, with already known descriptions of FracU(n+
0) as a Weyl skewfieldDn(n−1)/2(K) over a commutative field K which is a purely transcendental extension of dimen- sion n overC, and of FracU(b+
0) as a Weyl skewfieldDn(n+1)/2(C) over 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 superalgebraosp(1,2n) itself must be appreciated by recalling that even for the even part the answer is unknown, the case ofsp(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.
1. Analogues of Weyl algebras for polynomial superalgebras 1.1. Polynomial superalgebras.
1.1.1. Notations. It is usual in quantum group theory to denote byOΛ(Cn) the algebra of polynomials innindeterminatesy1, . . . , ynwith coefficients in Cand noncommutative multiplication twisted by relationsyiyj =λijyjyifor any 1≤i, j≤n, where Λ = (λij) is an×nmultiplicatively skew-symmetric matrix with entries in C. In the particular cases where all λij = 1, or λij =−1 for alli6=j, we use respectively the two following notations:
(i) O(Cn) is the commutative algebra of polynomials in n indetermi- natesy1, y2, . . . , yn with coefficients inC.
(ii) O(e Cn) is the noncommutative algebra of polynomials in nindeter- minatesy1, y2, . . . , yn with coefficients in C and product twisted by relationsyiyj =−yjyi for any 1≤i6=j≤n.
1.1.2. Superalgebra structure. The monomialsy1β1y2β2· · ·ynβn whose total degree β1+β2+· · ·+βn is even generate a commutative subalgebraR0 of O(Ce n). Denoting byR1theR0-submodule ofO(Ce n) generated byy1, . . . , yn, the C-submodules R0 and R1 satisfy O(e Cn) = R0 ⊕R1 and RiRj ⊆ Ri+j (with indices taken modulo 2). This gives rise to a structure of superalgebra onO(e Cn). The algebras O(e Cn) being noetherian domains, they admit skew fields of fractions. The following proposition gives an alternative form of O(e Cn) in terms of planesO(e C2) up to rational equivalence.
1.1.3. Proposition. For any integern≥2, we have:
Ifn= 2p, thenFrac (O(e Cn))'Frac (O(e C2)⊗p)
Ifn= 2p+ 1, then Frac (O(e Cn))'Frac (O(e C2)⊗p⊗ O(C))
Proof. Suppose thatn ≥3 and define from generators y1, . . . , yn of O(Ce n) with yiyj =−yjyi for all 1 ≤i 6= j ≤ n the monomials: y10 = y1, y20 = y2
and y0i=y1y2yi for any 3≤i≤n. They satisfyy10y20 =−y02y01 and y10yi0 =yi0y10, y20yi0 =yi0y20, yi0y0j =−y0jyi0 for all 3≤i6=j≤n.
It is clear that the subfield of FracO(e Cn) generated byy01, y20, . . . , y0nis equal to FracO(e Cn). We deduce that FracO(e Cn)'Frac (O(e C2)⊗O(e Cn−2)), 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 versions of quantum Weyl algebras. In the case of quantum spacesOΛ(Cn), these algebrasAq,Λn (introduced in [1] and [19] and studied in many papers) take in consideration both the quantization parameters Λ = (λij) of the
space and the quantization parametersq = (q1, . . . , qn) of the “differential”
operators on this space. We are interested here in two particular cases:
(i) Forλij =qi = 1 for all 1≤i, j≤n, we find the usual Weyl algebraAn(C), that is the algebra of polynomials in 2nindeterminatesx1, . . . , xn, y1, . . . , yn
with coefficients inCand a noncommutative product twisted by relations:
(xixj −xjxi =yiyj−yjyi =xiyj−yjxi = 0 (1≤i6=j ≤n),
xiyi−yixi = 1 (1≤i≤n). (1)
We also introduce the subalgebra Un(C) of An(C) generated by y1, . . . , yn
and w1, . . . , wn withwi=yixi for any 1≤i≤n. We have:
(wiwj−wjwi =yiyj−yjyi =wiyj−yjwi = 0 (1≤i6=j≤n),
wiyi−yiwi=yi (1≤i≤n). (2)
(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 indeterminatesx1, . . . , xn,y1, . . . , yn with coefficients in Cand a noncommutative product twisted by:
(xixj +xjxi =yiyj+yjyi =xiyj+yjxi = 0 (1≤i6=j ≤n),
xiyi−yixi = 1 (1≤i≤n). (3)
We denote by Uen(C) the subalgebra of Aen(C) generated by y1, . . . , yn and w1, . . . , wn withwi=yixi for any 1≤i≤n. We have:
yiyj+yjyi = 0 (1≤i6=j≤n), wiwj −wjwi =wiyj−yjwi = 0 (1≤i6=j≤n), wiyi−yiwi =yi (1≤i≤n).
(4)
1.2.2. Remarks. The algebras Aen(C) are the algebras denoted by Sn,nΛ 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 integern≥1, the algebra Aen(C) is simple, has center C, and has the same Hochschild homology and cohomology as the usual Weyl algebraAn(C).
1.2.3. Superalgebra structure.
(i) Letg0andg1 be twoC-vector spaces of dimensionnwith respective basis {z1, . . . , zn}and{y1, . . . , yn}. Letk0 be a linear extension ofg0of 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 3nby setting for the values of the super-bracket on these generators:
({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 constructiong0 is a subsuperalgebra ofk0 andg=g0⊕g1 is a subsuper- algebra ofk. It is clear by PBW theorem (see Theorem 6.1.2 in [14]) that the enveloping algebraU(k) is isomorphic toUen(C) of GK-dim 2n. Moreover the subalgebraU(k0) is isomorphic toUn(C) of GK-dim 2n, and the subalgebra U(g) is isomorphic to O(e Cn) of GK-dimn.
(ii) We have for the usual Weyl algebra the classical isomorphismAn(C)' A1(C)⊗n. We also have Aen(C)'Ae1(C)⊗nb in the tensor category of super- algebras assigning the parity 1 to the generatorsxi 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 FracAn(C) = FracUn(C) is the well known Weyl skewfield Dn(C), with center C. Simi- larly we deduce from (4) that FracAen(C) = FracUen(C), and it is proved in Proposition 3.3.1 of [18] that its center is alsoC. In parallel with Proposition 1.1.3, we have for these skewfields the following canonical form.
1.2.5. Theorem. For any integern≥1, we have the following isomorphisms:
(i) Ifn= 2pis even, then FracAen(C) = Frac (Ae2(C)⊗p).
(ii) Ifn= 2p+ 1is odd, thenFracAen(C) = Frac (Ae2(C)⊗p⊗A1(C)).
Proof. We can supposen≥1. We consider the subalgebraUen(C) defined in 1.2.1 with generatorsw1, . . . , wn, y1, . . . yn and relations (4). As in the proof of Proposition 1.1.3, we introduce: y10 =y1,y20 =y2 and yi0 =y1y2yi for 3≤ i≤n. Then we define: w01 =w1−(w3+· · ·+wn),w02 =w2−(w3+· · ·+wn) and w0i=wi for 3≤i≤n. Obvious calculations give:
y10y02=−y20y10, w01w20 =w02w01, y10w20 =w02y10, y02w10 =w10y20, [w01, y01] =y10, [w02, y20] =y20. (5) y10yj0 =y0jy10, y02yj0 =yj0y20, y01wj0 =w0jy10,
y20wj0 =wj0y02, w10w0j =wj0w10, w02wj0 =w0jw02, for 3≤j≤n. (6) We calculate for `= 1 or 2 and any 3 ≤j≤n the commutator: [w0`, y0j] = [w`, y1y2yj]−Pn
i=3[wi, y1y2yj] =y1y2yj−[wj, y1y2yj] = 0 to conclude : yj0w10 =w01yj0 and y0jw02 =w02y0j for any 3≤j≤n. (7) Moreover:
[wj0, yj0] =yj0 for any 3≤j≤n, (8) y0iy0j =−yj0y0i, w0iw0j =w0jwi0, w0iy0j =yj0wi0, for 3≤i6=j≤n. (9) We denote byV the subalgebra ofUen(C) generated byy01, . . . , yn0,w01, . . . , w0n, by W the subalgebra of V generated by y10, , y02, w01, w02, and by W0 the
subalgebra of V generated by y30, . . . , y0n, w30, . . . , wn0. It follows from (5) that W0 ' Ue2(C), from (8) and (9) that W00 ' Uen−2(C), and from (7) and (9) that V ' W ⊗W0. Since it is clear that FracV = FracUen(C), we deduce that FracUen(C) ' Frac (Ue2(C)⊗Uen−2(C)). We conclude that FracAen(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 partg1 of g=osp(1,2n) is a vector space of dimension 2nand the even partg0 is isomorphic to the symplectic Lie algebrasp(2n) of dimension 2n2+n. As a Lie superalgebra,osp(1,2n) is generated by the elements b±i (1 ≤ i ≤ n) of a basis of g1. Then the elements {b±j , b±k}(1≤j ≤k≤n) and {b+j , b−k} (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:
[{bξj, bηk}, b`] = (−ξ)δj`bηk+ (−η)δk`bξj (10) [{bξi, bηj},{bk, bϕ`}] = (−η)δjk{bξi, bϕ`}+ (−ξ)δik{bηj, bϕ`}
+ (ϕ−η)δj`{bξi, bk}+ (ϕ−ξ)δi`{bηj, bk}, (11) with 1≤i, j, k, `≤nand, ϕ, ξ, η are±. By PBW theorem, the enveloping algebraU(osp(1,2n)) is generated by the 2n2+nelements:
b±i and ki:= 12{b−i , b+i } for 1≤i≤n, (12) a±ij := 12{b±i , b±j } for 1≤i < j ≤n, (13) sij := 12{b−i , b+j } and tij := 12{b+i , b−j} for 1≤i < j ≤n, (14) of osp(1,2n). In particular, denoting
c±i := 12{b±i , b±i } for 1≤i≤n, (15) the enveloping algebra U(sp(2n)) of the even part g0 is the subalgebra of U(osp(1,2n)) generated by the 2n2 +n elements (b±i )2 = c±i , ki for 1 ≤ i ≤ n, and a±ij, sij, tij for 1 ≤ i < j ≤ n. The commutation relations in the associative algebraU(osp(1,2n)) are deduced from (10) and (11) taking {x, y} =xy+yx ifx, 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:
b+i b+j +b+j b+i = 2a+ij for all 1≤i < j≤n, (16) [c+i , b+j ] = [c+i , c+j ] = 0 for all 1≤i < j≤n, (17) [a+ij, b+` ] = [a+ij, c+`] = 0 for all 1≤i < j≤n, 1≤`≤n, (18) [a+ij, a+k`] = 0 for all 1≤i < j≤n, 1≤k < `≤n. (19)
Similarly with (10) and (14), we deduce for all 1≤i < j≤nand 1≤k≤n:
[tij, b+j ] =b+i and [tij, b+k] = 0 if k6=j. (20) The relations involving thetij’s are obtained from (11), (13) and (14), which give for 1≤i < j≤nand 1≤k < `≤n:
[tij, a+k`] =δjk1
2{b+i , b+` }+δj`1
2{b+i , b+k}, (21) [tij, tk`] =δjk 1
2{b+i , b−` } −δi`1
2{b+k, b−j}, (22) or in other words:
[tij, a+k`] = 0 if j6=k, j 6=` (1≤i < j ≤n, 1≤k < `≤n), (23) [tij, a+j`] =a+i` (1≤i < j < `≤n), (24)
[tij, a+ij] =c+i (1≤i < j ≤n), (25)
[tij, a+kj] =a+ik ifi < k (1≤i < j≤n, 1≤k < j≤n,), (26) [tij, a+kj] =a+ki ifk < 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 ifj6=k, i6=` (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, b+i ] =b+i , [ki, b±j ] = 0 (1≤i6=j≤n), (32) then with (13) and (14), for all 1≤i < j≤nand 1≤`≤n:
[ki, a+ij] =a+ij, [kj, a+ij] =a+ij,
[k`, a+ij] = 0 if`6=i, j.
[ki, tij] =tij, [kj, tij] =−tij, [k`, tij] = 0 if`6=i, j.
(33)
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:=b+n, (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] =b+n−1. (37)
It also follows from (20) that for any 1≤i≤n−1 and 1≤j≤n:
[ti, b+i+1] =b+i and [ti, b+j] = 0 if j6=i+ 1. (38)
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]:
t2iti+1−2titi+1ti+ti+1t2i = 0 (1≤i≤n−1), (39) t2iti−1−2titi−1ti+ti−1t2i = 0 (2≤i≤n−1), (40) t3ntn−1−(t2ntn−1tn+tntn−1t2n) +tn−1t3n= 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 subsuperalge- bra of osp(1,2n) generated by Chevalley generatorst1, t2, . . . , tn. By (34), (37) and (38), n+ contains the elements b+1, b+2, . . . , b+n. Hence by (13), n+ contains the elements a+ij for 1≤i < j≤n. It also follows inductively from (28) and (36) thatn+ contains the elementstij for 1≤i < j≤n. Hence:
n+=n+
0 ⊕n+
1, where n+
1 is the C-vector space with basis (b+i )1≤i≤n, and n+
0 'sp+(2n) is the Lie subalgebra ofg0'sp(2n) with basis (c+i , a+ij, tij)1≤i<j≤n. Then:
dimn+
0 = dimsp+(2n) =n2 and dimn+
1 =n.
Let us observe that relations (17), (18) and (19) imply that the Lie subal- gebra with basis (c+i , a+ij)1≤i<j≤n inn+
0 is abelian of dimensionn(n+ 1)/2.
2.3.2. Description of U(n+) as an iterated skew polynomial algebra. We de- note by U+ the enveloping algebra U(n+) of n+. By PBW theorem U+ is generated as an associative algebra overCby then2 elements b+i ,a+ij andtij for 1≤i < j ≤n. The relations detailed in 2.1 allow us to describe U+ as an iterated skew polynomial extension. We start with A:= C[a+ij]1≤i<j≤n, which is by (19) a commutative polynomial algebra inn(n−1)/2 variables.
Then we set:
B0 :=A, B1 :=A[b+1], Bj :=Bj−1[b+j ;σj, δj] for 2≤j≤n, (42) where σj is the automorphism of Bj−1 and δj is the σj-derivation of Bj−1
defined by σj(a+k`) = a+k` and δj(a+k`) = 0 for all 1 ≤ k < ` ≤ n, and σj(b+i ) = −b+i and δj(b+i ) = 2a+ij for any 1 ≤ i < j, in order to translate relations (16) and (18). At the last step we denote:
B :=Bn=C[a+ij][b+1][b+2 ;σ2, δ2]· · ·[b+n;σn, δn]. (43) Then we add the generatorstij 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+,
where the derivations dij are suitably defined to account for relations (20) and (23) to (30). To sum up we conclude:
A=C[a+ij]1≤i<j<n, (44) B =A[b+i ;σi, δi]1≤i≤n, (45) U+=B[tij;dij]1≤i<j≤n. (46) Let us recall that in this presentation, A is commutative and theb+i ’s com- mute with the elements ofA.
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 elements c+k = (b+k)2 for 1≤k≤n. It follows from previous results that:
C =C[a+ij, c+k]1≤i<j<n,1≤k≤n, (47) U(n+0) =C[tij;dij]1≤i<j≤n. (48) 2.4. Enveloping algebra of the solvable Borel subsuperalgebra b+. 2.4.1. Definition of b+. The Cartan Lie subalgebra hof g0 is the Lie subal- gebra 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 byn+ and h. We have clearly:
b+=n+⊕h=b+
0 ⊕n+
1, with b+
0 :=n+
0 ⊕h, where the even partb+
0 is the positive Borel Lie subalgebra ofg0'sp(2n).
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 ofUb+ 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(b+i ) = b+i and
∆i(b+j ) = 0 for any 1 ≤ j 6= i ≤ n, ∆i(a+ij) = a+ij and ∆i(tij) = tij for 1 ≤ i < j ≤ n, ∆i(a+ji) = a+ji and ∆i(tji) = −tji for 1 ≤ j < i ≤ n,
∆i(a+k`) = 0 and ∆i(tk`) = 0 for k6=i, `6=i.
2.4.3. Restriction to the even part. It follows from previous results that, with notations (49) and (48):
U(b+0) =U(n+0)[ki; ∆i]1≤i≤n=C[tij;dij]1≤i<j≤n[ki; ∆i]1≤i≤n. (50) 3. Proof of the main theorem
3.1. Some general technical results. We will use the following easy lem- mas.
3.1.1. Lemma. LetAbe a commutativeC-algebra,aan element ofA, andR the iterated skew polynomial algebra generated overAby two generatorsx1 and x2 commuting with the elements ofA and with relationx1x2+x2x1= 2a. Then the elementy=x1x2−asatisfies x1y=−yx1 and x2y=−yx2.
Proof. Obvious calculation.
3.1.2. Lemma. Let A be a commutative C-algebra. Let (aij)1≤i<j≤n be a family of elements of A, and R the iterated skew polynomial algebra gen- erated over A by n generators x1, x2, . . . , xn commuting with the elements of A and with relations xixj +xjxi = 2aij for all1 ≤i < j ≤n. Then the n−1elements yi =x1xi−a1i for any 2≤i≤nsatisfy:
x1yi=−yix1 and yiyj+yjyi = 2(a1ia1j −aijx21).
Proof. The first identity follows from previous lemma. For the second one, we observe firstly that:
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
=−yix1xj−2aijx21+ 2a1jx1xi−a1jyi
=−yi(x1xj −a1j)−2a1jyi−2aijx21+ 2a1jx1xi
=−yiyj−2a1jyi−2aijx21+ 2a1jx1xi
=−yiyj+ 2a1j(x1xi−yi)−2aijx21.
We conclude that yjyi=−yiyj+ 2a1ja1i−2aijx21. 3.2. Enveloping skewfield of the Lie subsuperalgebra n+.
3.2.1. Notations. Recalling the description (44), (45), (46) ofU+, we intro- duce the skewfields of rational functions:
K:= FracA=C(a+ij)1≤i<j≤n, (51) L:= FracB =K(b+i ;σi, δi)1≤i≤n, (52) F := FracU+=L(tij;dij)1≤i<j≤n. (53) 3.2.2. Notations. We define the following families of elements ofB:
y1:=b+1, x1,i:=b+i , u1,i,j :=a+i,j for all 1≤i < j≤n, (54) and more generally for any 1≤k≤n:
uk,i,j :=uk−1,k−1,iuk−1,k−1,j−uk−1,i,jyk−12 for all k≤i < j ≤n, xk,i:=xk−1,k−1xk−1,i−uk−1,k−1,i for any k≤i≤n, yk:=xk,k.
(55)
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 elementsy1, y2, . . . , yn ofB satisfy the relations:
yk−1yk=−ykyk−1 for any 2≤k≤n,
yky` =y`yk for all 1≤k, `≤n, `6=k±1, yka+ij =a+ijyk for all 1≤k≤n,1≤i < j≤n.
(ii) For any1≤k≤n, the elements uk,i,j withk≤i < j ≤nare in the commutative subalgebraA[y12, y22, . . . , yk−12 ]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 allk≤i < j ≤n. (58) We deduce by induction on `that, for any 1≤`≤n:
y`x`+1,i=−x`+1,iy` for any `+ 1≤i≤n, y`xk,i=xk,iy` for any `+ 2≤k≤i≤n, y`uk,i,j =uk,i,jy` for any 1≤k≤i < j ≤n.
(59) Point (i) of the lemma follows obviously from these relations and point (ii) follows by iterative definition (55) of theuk,i,j’s.
3.2.4. Proposition.
(i) The subalgebra ofB generated over Cby y1, . . . , yn is isomophic to OΛ(Cn), where all entries of the symmetric matrix Λ = (λij)1≤i,j≤n
are equal to 1 exceptλi,i+1=−1 =λi+1,i for any 1≤i≤n−1.
(ii) Its skewfield of fractionsQ satisfies:
L=Q(a+ij)1≤i<j≤n, (60)
Q'Frac (O(e Cn)), (61)
where eacha+ij commutes with the elements of Q.
Proof. By induction from relations (55), we prove that for any 1≤k≤n, yk= (yk−1yk−2· · ·y2y1)b+k +rk−1, (62) where the termrk−1lies in the subalgebra ofBk−1generated byy1, . . . , yk−1
and the a+ij’s for 1≤ i < j ≤ k. It follows that the skewfield L defined by (52) is also the skewfield of rational functions generated overKbyy1, . . . , yn. Then the commutation relations of Lemma 3.2.3 imply point (i) and asser- tion (60) of point (ii). In order to prove assertion (61), we suppose that n≥3 and define: y10 =y1,y02 =y2,y03=y1y3 and y0i=yi for any 4≤i≤n.
It is clear that the subfield ofQgenerated by the elements (y0i)1≤i≤nis equal toQ. Moreover by straightforward calculations, we have:
y10y20 =−y20y10, y01yi0 =yi0y10, y02y0i=y02y01, for any 3≤i≤n, y30y40 =−y40y30, y03yi0 =yi0y30, for any 5≤i≤n, We deduce that Q'Frac (O(Ce 2)⊗Tn−2) where Tn−2 is the subalgebra of Q generated by the elements (y0i)3≤i≤n. Applying the same reduction on the generators ofTn−2, we conclude by iteration and using Proposition 1.1.3
thatQ'Frac (O(Ce n)).
3.2.5. Lemma. For any1≤i≤n−1 and any 1≤m≤n, we have:
[ti, um,i,i+1] =x2m,i (63)
[ti, um,k,i+1] =um,k,i form≤k < i≤n, (64) [ti, um,i+1,`] =um,i,` form < i+ 1< `≤n, (65)
[ti, um,i+1,`] = 0 form=i+ 1< `≤n, (66)
[ti, um,k,`] = 0 fork6=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 fork6=i+ 1. (70) Proof. We study the action onB of the Chevalley generatorst1, t2, . . . , tn−1
defined in 2.2. Let us recall that, by (38) we have:
[ti, b+i+1] =b+i and [ti, b+k] = 0 ifk6=i+ 1. (71) Similarly by (23) to (27), we obtain:
[ti, a+i,i+1] = (b+i )2, (72)
[ti, a+k,i+1] =a+k,i for 1≤k < i≤n, (73) [ti, a+i+1,`] =a+i,` fori+ 1< `≤n, (74) [ti, a+k,`] = 0 fork6=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 casem= 1 of assertions (68) to (70), and (72) to (75) are the casem= 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,`]
−[ti, um,k,`]x2m,m−um,k,`[ti, x2m,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 rankm+ 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
m+ 1 = i+ 1 and equals xm+1,i if i+ 1 > m+ 1, and [ti, xm+1,k] = 0 for
any k6=i+ 1. This finishes the proof.
3.2.6. Corollary. Each generator tij for1≤i < j≤ncommutes with each generator ym for1≤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 ≤nis an iterated bracket of t1, t2, . . . , tn−1. 3.2.7. Proposition. The skewfield of fractions ofU+ satisfies:
F =Q(a+ij)1≤i<j≤n(tij;dij)1≤i<j≤n, (76) where the generatorstij commute with all elements ofQ, and the derivations dij translate the commutation relations between the tij’s and the a+ij’s [see relations(23)to(27)] or thetij’s with each other [see relations(28)to(30)].
Proof. This is just a summary of assertions (53) and (60), Proposition 3.2.4
and Corollary 3.2.6.
3.2.8. Lemma. Denote by (qi,j)1≤i<j≤n the elements of F defined by:
qi,j :=yi−2ui,i,j for any 1≤i < j ≤n. (77) Then the qi,j’s commute with each other and with all elements of Q, and the subfield of L generated overQ by the(qi,j)1≤i<j≤n is equal to L.
Proof. The first assertion follows from Corollary 3.2.3 and third relation in (59). To prove the second assertion, we observe directly from definition (55) that for any 1≤i < j ≤n, we have:
ui,i,j=ei−1,j+ (−1)i−1a+i,jy12y22· · ·yi−12 ,
where the elementei−1,jlies in the subalgebraEi−1generated byy12, y22, . . . , yi−12 and thea+k,`’s for 1≤k≤i−1. Then we can write:
a+i,j = (−1)i−1y1−2y−22 · · ·y−2i−1qi,j+e0i−1,j
withe0i−1,j := (−1)iy−21 y2−2· · ·yi−1−2ei−1,j∈Ei−1. This proves the rest of the
lemma by induction on i.
3.2.9. Lemma. For any1≤i < j ≤nand any 1≤k < `≤n, we have:
[tij, qi,j] = 1, (78)
[tij, qk,`] = 0 ifj 6=`, (79) [tij, qk,j] = 0 ifi < k, (80) [tij, qk,j] =qk,i ifi > 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) thatti,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
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<j≤n defined in Lemma 3.2.8 and the elements(pi,j)1≤i<j≤n defined inductively by:
p1,j :=t1,j and pi,j :=ti,j−Pi−1
m=1qm,ipm,j (82)
satisfy[pij, qij] = 1and [pij, pk`] = [qij, qk`] = [pij, qk`] = 0for(i, j)6= (k, `).
Proof. Follows obviously from Lemma 3.2.9 by induction oni.
3.2.11. Proposition. We have: F =Q(qij)1≤i<j≤n(pij;∂qij)1≤i<j≤n.
Proof. It’s clear by Lemma 3.2.8 thatL=Q(qij)1≤i<j≤n. By definition (82) and Proposition 3.2.7, the subfield of F generated over L by the elements (pi,j)1≤i<j≤nis equal toF. Hence the proposition just summaries the results
of Lemma 3.2.8 and Corollary 3.2.10.
3.2.12. Corollary. The skewfieldFis isomorphic toFrac (An(n−1)/2⊗O(Ce n)), and its center is Z(F) =Z(Q) =C(y12, . . . , yn2).
Proof. By Corollary 3.2.10, the generators pij and qij commute with all elements ofQand the subalgebraW ofF 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 subalgebran+
0.
3.3.1. Notations. In order to determine the skewfield of fractions of the en- veloping algebra of the even part n+
0 of n+ described in (48), we complete the notations (51), (52), (53) and (61) by defining:
Q0:=C(y12, y22, . . . , yn2) ⊂ Q, (83) L0:=K((b+1)2,(b+2)2, . . . ,(b+n)2) ⊂ L, (84) F0:= FracU(n+0) =L0(tij;dij)1≤i<j≤n ⊂ F. (85) 3.3.2. Lemma. We have the equalityL0 =K(y21, y22, . . . , y2n).
Proof. Returning to notations (55) and using (59) and (58), we check by straightforward calculations that: x2k,i=−x2k−1,k−1x2k−1,i+u2k−1,k−1,ifor all 2≤k≤i≤n. Using this identity and point (ii) of Lemma 3.2.3, we prove thaty12 = (b+1)2,y22 =−(b+1)2(b+2)2+a21,2, and :
yi2 = (−1)i−1y12y22· · ·y2i−1(b+i )2+vi−1 for any 2≤i≤n,
where the termvi−1lies in the subalgebra generated overAbyy12, y22, . . . , yi−12 .
Then the result follows by induction oni.
