Coideal algebras from twisted Manin triples

TL;DR

This paper introduces Lie bi-ideal structures derived from automorphisms of Manin triples, providing a new framework to generate coideal algebras, including twisted Yangians and q-Onsager algebras, with new presentations and insights.

Contribution

It presents a novel approach using automorphisms of Manin triples to define Lie bi-ideal structures that deform into coideal algebras, unifying various examples.

Findings

Recovered twisted Yangians from the new framework

Derived q-Onsager algebra and augmented q-Onsager algebra

Found a new presentation for twisted Yangians

Abstract

We propose a new approach to study coideal algebras. It is well-known that Manin triples (or equivalently Lie bi-algebra structures) are the requirement to deform Lie algebras and to obtain quantum groups. In this paper, introducing some particular automorphisms of Manin triples, we define new structures that we call Lie bi-ideal structures. A link with coisotropic subalgebras is explained. We show that their deformation provide coideal algebras. As examples, we recover from our general construction the twisted Yangians, the q-Onsager algebra and the augmented q-Onsager algebra. As an important by-product, we find a new presentation for the twisted Yangians.