On shifted super Yangians and a class of finite W-superalgebras

Abstract

Abstract. Let e be an even nilpotent element, satisfying certain restrictions on its Jordan type, in a general linear Lie superalgebra. We study the finite W -superalgebra W e associated to such an e, and a realization of W e in terms of a quotient of a shifted super Yangian is established.

On Kac–Weisfeiler modules for general and special linear lie superalgebras

“…In this section we will first recall some material on the pyramids for g := gl m|n . One can refer to the references [1], [2], [3] and [7] for more details on pyramids. We will then introduce the so-called "shifted pyramids" associated to the Dynkin pyramids of gl m|n which will be crucially useful to the sequent arguments.…”

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

“…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].…”

Finite W-Superalgebras and Truncated Super Yangians

“…The very first result is obtained by Briot and Ragoucy [BR], saying that if the nilpotent element e ∈ gl M |N is rectangular, then the associated finite W -superalgebra is isomorphic to a certain quotient of Y m|n called the truncated super Yangian, where m and n are the numbers of Jordan blocks of e restricted to the even and odd spaces, respectively. In recent years, there have been some results [BBG,Pe2,Pe3] generalizing the above observation when the nilpotent element e satisfies some assumptions, but for a general e the problem remains to be open.…”

“…Similar to the Lie algebra case [BK1,Dr2], the RTT presentation and the Drinfeld's presentation can be treated as special cases of the parabolic presentations. There have been some results [Go,Pe1] giving suitable presentations of Y m|n , where the results [BBG,Pe3] are in fact based on them. However, as noticed in [BBG,Pe3], they are no longer suitable presentations for the general case.…”

“…There have been some results [Go,Pe1] giving suitable presentations of Y m|n , where the results [BBG,Pe3] are in fact based on them. However, as noticed in [BBG,Pe3], they are no longer suitable presentations for the general case. What we need is a kind of "more" generalized parabolic presentation which works for any 01-sequence [CW, FSS], which is a parametrizing set controlling the parities of elements in Y m|n .…”