Twisted q-Yangians and Sklyanin determinants

Abstract

$q$-Yangians can be viewed as quantum deformations of the upper triangular loop Lie algebras, and also be viewed as deformation of the Yangian algebra. In this paper, we study the twisted $q$-Yangians as coideal subalgebras of the quantum affine algebra introduced by Molev, Ragoucy and Sorba. We investigate the invariant theory of the quantum symmetric spaces in affine types $AI, AII$ and use the Sklyanin determinants to study the invariant theory and show that they also obey classical type identities similar to the quantum coordinate algebras of finite types.