Sommaire:

We will consider the quantification of the invariant sub-algebras of the loop and of the half-loop Lie algebras of sl(2). This quantification use the notion of twist for Manin triples. The Manin triples are usually used in the construction of the quantum loop and half-loop Lie algebra of sl(2). From this construction, we obtain four co-ideal algebras: the q-Onsager and the augmented q-Onsager for the " loop" case and the two twisted Yangians for the "half-loop" case. We will discuss the role of these algebras in the context of quantum integrable models with boundaries and reflection equations. Some open problems about the "Drinfel'd second realisation" of the reflection algebra and the Q baxter operators will be presented.

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