Quantum Berezinian for the Twisted Super Yangian
Hongda Lin, Yongjie Wang, Honglian Zhang
TL;DR
The paper resolves open questions on the quantum Berezinian for twisted super Yangians by providing an explicit construction of $ rak{B}^{tw}_{M|N}(u)$, establishing its centrality, and defining the special twisted Yangian as a center-quotient. It develops an equivalent presentation for $ ext{Y}^{tw}( rak{osp}_{M|N})$, analyzes the center via a twisted polynomial current algebra, and proves a quantum Liouville formula linking the center to the Berezinian. A Sylvester-type theorem is then formulated for the extended twisted super Yangian, using quasi-determinants to relate the Berezinian across extensions. Collectively, these results illuminate central, structural, and representational aspects of twisted super Yangians and provide concrete computational tools for their quantum invariants and Sylvester relations, with potential applications to representations and Harish-Chandra-type theories in the super setting.
Abstract
Motivated by an open problem proposed in Molev's book \cite[Section 2.16, Example 16]{Mo07}, we investigate the quantum Berezinian $\mathfrak{B}^{tw}(u)$ associated with the twisted super Yangian, which is a coideal sub-superalgebra of the super Yangian of the general linear Lie superalgebra. We provide an explicit formulation of $\mathfrak{B}^{tw}(u)$, and we also construct the center of the twisted super Yangian. This construction enables us to define the special twisted super Yangian, which is isomorphic to the quotient of the twisted super Yangian by its center. Moreover, we demonstrate the quantum Sylvester theorem for both the generator matrix and the quantum Berezinian.