Representations of the super-Yangian of type $D(n,m)$

TL;DR

This paper studies finite-dimensional irreducible representations of Yangians related to orthosymplectic Lie superalgebras, providing necessary conditions, conjecturing sufficiency, and proving it for certain cases using new odd reflection techniques.

Contribution

It introduces necessary conditions for finite-dimensionality, conjectures their sufficiency, and proves the conjecture for a class of linear highest weight representations, also constructing an isomorphism between related Yangians.

Findings

Necessary conditions for finite-dimensional irreducible representations.

Conjecture on sufficiency of these conditions.

Proof of the conjecture for linear highest weight cases.

Abstract

We consider the classification problem for finite-dimensional irreducible representations of the Yangians associated with the orthosymplectic Lie superalgebras ${\frak{osp}}_{2n|2m}$ with $n\geqslant 2$. We give necessary conditions for an irreducible highest weight representation to be finite-dimensional. We conjecture that these conditions are also sufficient and prove the conjecture for a class of representations with linear highest weights. The arguments are based on a new type of odd reflections for the Yangian associated with ${\frak{osp}}_{2|2}$. In the Appendix, we construct an isomorphism between the Yangians associated with the Lie superalgebras ${\frak{osp}}_{2|2}$ and ${\frak{gl}}_{1|2}$.