A Minimal Representation of the Orthosymplectic Lie Supergroup

We construct a minimal representation of the orthosymplectic Lie supergroup $OSp(p,q|2n)$ for $p+q$ even, generalizing the Schrödinger model of the minimal representation of $O(p,q)$ to the super case. The underlying Lie algebra representation is realized on functions on the minimal orbit inside the Jordan superalgebra associated with $\mathfrak{osp}(p,q|2n)$, so that our construction is in line with the orbit philosophy. Its annihilator is given by a Joseph-like ideal for $\mathfrak{osp}(p,q|2n)$, and therefore the representation is a natural generalization of a minimal representation to the context of Lie superalgebras. We also calculate its Gelfand–Kirillov dimension and construct a nondegenerate sesquilinear form for which the representation is skew-symmetric and which is the analogue of an $L^2$-inner product in the supercase.

JORDAN BIMODULES OVER THE SUPERALGEBRA M1|1

Let F be a field of characteristic different of 2 and let M1|1(F)(+) denote the Jordan superalgebra of 2 × 2 matrices over the field F. The aim of this paper is to classify irreducible (unital and one-sided) Jordan bimodules over the Jordan superalgebra M1|1(F)(+).

A RELATION BETWEEN SUPERTRIALITY AND SUPERTWISTOR

We write down our supertriality axioms and introduce the Jordan superalgebra associated to a supertriality. We show how to write the supertriality operations directly in terms of supertwistors.

IRREDUCIBLE REPRESENTATIONS OF A CERTAIN JORDAN SUPERALGEBRA

A classification of unital finite-dimensional irreducible Jordan representations of the simple superalgebra D(t) in the case of characteristic 0 was obtained. As a corollary we obtain a classification of finite-dimensional irreducible representations of the Kaplansky superalgebra K3 in the case of characteristic 0.