Quantum Berezinian for the Twisted Super Yangian

TL;DR

This paper explicitly formulates the quantum Berezinian for the twisted super Yangian, constructs its center, and establishes the quantum Sylvester theorem, advancing understanding of its algebraic structure.

Contribution

It provides an explicit formulation of the quantum Berezinian and constructs the center of the twisted super Yangian, introducing the concept of the special twisted super Yangian.

Findings

Explicit formulation of the quantum Berezinian for twisted super Yangian

Construction of the center of the twisted super Yangian

Proof of the quantum Sylvester theorem for the generator matrix and Berezinian

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.