The Joseph Ideal for $$\mathfrak {sl}(m|n)$$

Abstract

Using deformation theory, Braverman and Joseph obtained an alternative characterisation of the Joseph ideal for simple Lie algebras, which included even type A. In this note we extend that characterisation to define a remarkable quadratic ideal for sl(m|n). When m − n > 2 we prove the ideal is primitive and can also be characterised similarly to the construction of the Joseph ideal by Garfinkle.

Super-Laplacians and their symmetries

“…The above discussion can be taken over to the supersymmetric case. It was given for osp and spo in [16] and for sl in [35]. It turns out that the special tensor argument gives the same answers provided that traces are replaced by supertraces, although there are some special values of the dimensions of the algebras for which the discussion does not hold.…”

“…The product of two differential operators on super-twistor space has the value of λ one would expect by taking the purely even case and replacing the trace by the supertrace, and this is also what one finds from the earlier superspace discussions. However, the special tensor argument of [35] breaks down for N = 2, 4, 6. The cases of most interest from a physical point of view are N = 1, 2, 3, 4, so that this result is rather unfortunate especially for analytic superspace.…”