On Kostant's theorem for the Lie superalgebra Q(n)

Abstract

A finite W -algebra is a certain associative algebra attached to a pair (g, e) where g is a complex semisimple Lie algebra and e ∈ g is a nilpotent element. Geometrically a finite W algebra is a quantization of the Poisson structure on the so-called Slodowy slice (a transversal slice to the orbit of e in the adjoint representation). In the case when e = 0 the finite W -algebra coincides with the universal enveloping algebra U(g) and in the case when e is a regular nilpotent element, the corresponding W -algebra coincides with the center of U(g). The latter case was studied by B. Kostant [15] who was motivated by applications to generalized Toda lattices. The general definition of a finite W -algebra was given by A. Premet in [24]. I. Losev used the machinery of Fedosov quantization to prove important results relating representations of Walgebras and primitive ideals of U(g) [16,17,18] (see also [25,26,27]). He used this result to prove long standing conjectures of A. Joseph and others concerning primitive ideals in U(g), [11].On the other hand, affine W -algebras were first constructed by physicists [8,9]. The role of the Slodowy slice in W -algebras in the principal case was recognized in [2]. A. De Sole and V.G. Kac in [7] established the relation between affine and finite W -algebras.Let us mention an important discovery of physicists, [28], that for g = sl(n) finite W -algebras are closely related to Yangians. This connection was further studied in [4] and [6].It is interesting to generalize all above applications to Lie superalgebras. Finite Walgebras for Lie superalgebras have been extensively studied by C. Briot, E. Ragoucy, J. Brundan, J. Brown, S. Goodwin, W. Wang, L. Zhao and other mathematicians and physicists [3,5,31,32]. Analogues of finite W -algebras for Lie superalgebras in terms of BRST cohomology were defined in by A. De Sole and V.G. Kac in [7].In [3] C. Briot and E. Ragoucy observed that finite W -algebras associated with certain nilpotent orbits in gl(pm|pn) can be realized as truncations of the super-Yangian of gl(m|n), see [19] for definition.The principal finite W -algebras for gl(m|n) associated to regular (principal) nilpotent elements were described as certain truncations of a shifted version of the super-Yangian Y (gl(1|1)) in [5]. It is also proven there that all irreducible modules over 1

On the finite W-algebra for the Lie superalgebra Q(N) in the non-regular case

“…In [13] it is proven for D(2, 1; α) and a regular nilpotent χ. We proved this conjecture for g = Q(n) in the regular case in [12] (Corollary 4.9). It follows from our results, that the conjecture is also true in the case that we consider in this paper (Corollary 5.5).…”

“…Remark 4.3. In [12] we proved this theorem in the case when e is a regular even nilpotent element, i.e. l = n.…”

“…The general definition of a finite W -algebra was given by A. Premet in [15] (see also [7]). In the case of Lie superalgebras, finite W -algebras have been extensively studied by mathematicians and physicists in [1,2,[10][11][12][13][14][18][19][20].…”

“…In [12] we studied in detail Q(n) in the regular case. In particular, we proved that W χ for Q(n) associated to a regular nilpotent χ is isomorphic to a quotient of the super-Yangian of Q(1) (Theorem 6.2).…”

“…In [12] we formulated the following conjecture (Conjecture 2.8): Assume that g is a Lie superalgebra with a reductive even part g0 endowed with a Z-grading g = ⊕ j∈Z g j , which is good for χ. Then if dim(g −1 )1 is even, we have that Gr K W χ ≃ S(g χ ) and if dim(g −1 )1 is odd, then…”