Finite W-superalgebras W(e) are certain associative superalgebras, determined by an even nilpotent element e in the general linear Lie superalgebra gl(M|N), up to isomorphism. It is proved that the algebraic structure of W(e) can be realized as a quotient of a certain subalgebra of the Yangian Y(m|n) associated to gl(m|n), where m and n are determined by the Jordan canonical form of e. As an application, one may expect to adapt the approaches used to the study of representation theory of Yangians to that of W-superalgebras.

 

In this talk, we will focus on the special case where e is principal nilpotent. In this case, a complete classification of irreducible modules of W(e) was obtained by Brown-Brundan-Goodwin [2013]. The proof is based on a triangular decomposition and a PBW basis theorem of W(e), which in turn allows one to define the Verma module and hence establish its highest weight theory similar to the case of Lie superalgebras. We expect a careful study of this special case will not only provide useful information toward the general e case (which remains to be open), but also provide new insights to the study of Whittaker modules over gl(m|n).