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

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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 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 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 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- 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 FRANC¸OIS 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 defi- nition 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 al- gebra 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 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- 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 corol- lary 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 dimen- sion 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 indetermi- nates y1; y2; : : : ; yn with coefficients in C. n (ii) Oe(C ) is the noncommutative algebra of polynomials in n indeter- minates 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.
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