Some Lie superalgebras associated to the Weyl algebras

Abstract

Abstract. Let g be the Lie superalgebra osp(1, 2r). We show that there is a surjective homomorphism from U (g) to the r th Weyl algebra Ar, and we use this to construct an analog of the Joseph ideal. We also obtain a decomposition of the adjoint representation of g on Ar and use this to show that if Ar is made into a Lie superalgebra using its natural Z 2 -grading, then We work throughout over an algebraically closed field k of characteristic zero. If g is a simple Lie algebra different from s (n), Joseph shows in [J2], that there is a unique completely prime ideal, J 0 whose associated variety is the closure of the minimal nilpotent orbit in g * . When g is the symplectic algebra g = sp(2r), this ideal may be constructed as follows. It is well known that the symmetric elements of degree two in the r th Weyl algebra A r form a Lie algebra isomorphic to sp(2r) [D, Lemma 4.6.9]. Hence there is an algebra map φ : U (g) −→ A r whose kernel is clearly completely prime and primitive. Since the image of φ has Gel'fand Kirillov dimension 2r, and this is the dimension of the minimal nilpotent orbit in g * by [CM, Lemma 4.3.5], we have ker φ = J 0 . Now if g is a classical simple Lie superalgebra, and U (g) contains a completely prime primitive ideal different from the augmentation ideal, then g is isomorphic to an orthosymplectic algebra osp(1, 2r) (Lemma 1). We observe that if g = osp(1, 2r), then there is a surjective homomorphism U (g) −→ A r whose kernel J satisfies J ∩ U (g 0 ) = J 0 . The existence of this homomorphism has previously been shown in the Physics literature; see, for example, [F, pages 55 and 170]. It follows that g acts via the adjoint representation on A r , and we determine the decomposition of this representation explicitly.This turns out to be a useful setting in which to study the Lie structure of certain associative algebras. A result of Herstein [He]

Joseph Ideals and Harmonic Analysis for

“…For a classical simple Lie superalgebra g, the universal enveloping algebra U(g) never contains a completely prime primitive ideal, different from the augmentation ideal, unless g is isomorphic to the orthosymplectic Lie superalgebra osp(1|2n), see [Mu99]. In spite of the geometric origin of osp(m|2n) as a conformal Lie superalgebra on R m−2|2n , the previous argument implies that there is no direct counterpart of the Joseph ideal of classical Lie algebras.…”

Supertrace and Superquadratic Lie Structure on the Weyl Algebra, and Applications to Formal Inverse Weyl Transform

“…Let us quote that the simplicity of [W, W] was known some time ago, as a combination of a result by S. Montgomery [12] proving that [W, W]/K is simple and a result by I. M. Musson [13] proving that W = K ⊕ [W, W]. Our proof using a supertrace and the Moyal product is direct and completely different of the initial proofs in [12] and [13].…”

“…[7]) and it was used for instance to develop singleton Anti-de-Sitter theories [7]. This embedding was also described in [13] and [8]. We give the proof in [8]: Definition 2.2.…”

“…Briefly, in a deformation quantization framework (see [3,17]), we quantize the natural Poisson bracket in an explicit way (Moyal ⋆-product). The main advantage of this quantization is the fact of being invariant with respect to the natural embedding of the Lie superalgebra osp (1, 2n) in W ( [7,13,8]), i.e. with respect to the orthosymplectic supersymmetry.…”

Lie superalgebras, Clifford algebras, induced modules and nilpotent orbits

“…For example they are important in the determination of the scale factor in Goldie rank polynomials, and they are related to unitary representations, see [J3] for more details. On the other hand if g is a classical simple Lie superalgebra, there are very few completely prime ideals in U(g), see [M3,Lemma 1].…”