Representations of the super-Yangian of type $D(n,m)$
A. I. Molev
TL;DR
The work addresses the finite-dimensional irreducible representations of the Yangian $Y(osp_{2n|2m})$ for $n\ge 2$ by developing a framework that combines RTT-based extended Yangians, parity-sequence analysis, and a novel odd-reflection mechanism for $osp_{2|2}$. It derives necessary conditions for finite-dimensionality, links them to an $(m,n)$-hook diagram parametrization, and proves sufficiency in the special case of linear highest weights, aided by a vector-representation tensor-product construction and embedding results. A key technical contribution is the explicit odd-reflection transformation of highest weights and the associated fdimm criterion involving a monic polynomial $P(u)$; the Appendix also establishes an isomorphism between $Y(osp_{2|2})$ and $Y(gl_{1|2})$, enriching the structural bridge between the algebras. Overall, the paper advances a conjectural classification by integrating reflection techniques, rank-reduction embeddings, and constructive representation-building to obtain concrete necessary-and-sufficient conditions for finite-dimensional irreducible modules.
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 ${\frak{gl}}_{1|2}$.