Twisted q-Yangians and Sklyanin determinants

TL;DR

This paper explores the structure of twisted q-Yangians and quantum affine algebras, introducing the twisted Sklyanin determinant and establishing key identities to advance understanding of their invariant theory.

Contribution

It introduces the twisted Sklyanin determinant for twisted quantum affine algebras and derives fundamental identities, expanding the algebraic framework of quantum deformations.

Findings

Defined the twisted Sklyanin determinant.

Established identities for the Sklyanin determinants.

Analyzed the invariant theory of twisted quantum affine algebras.

Abstract

$q$-Yangians can be viewed both as quantum deformations of the loop algebras of upper triangular Lie algebras and deformations of the Yangian algebras. In this paper, we study the quantum affine algebra as a product of two copies of the $q$-Yangian algebras. This viewpoint enables us to investigate the invariant theory of quantum affine algebras and their twisted versions. We introduce the twisted Sklyanin determinant for twisted quantum affine algebras and establish various identities for the Sklyanin determinants.