FRT presentation of the q-Onsager comodule algebra and refined q-characters

We introduce a concept of counital comodule algebra (B,δB,εB) over a Hopf K-algebra H, where δB: B→ B ⊗ H and εB: B→ K are coaction and counit homomorphisms, respectively. This structure gives a canonical (counital) comodule algebra map ιB: B → H. In this paper, we study in details the case B=Oq, and H is the quantum loop algebra LUqsl₂ for K= C. Here, Oq is the q-Onsager algebra with four presentations: (i) the q-Dolan-Grady type; (ii) the alternating type; (iii) the root vectors type; (iv) the Drinfeld type. For (Oq,δOq,εOq), in fact a 3-parameter family of counital comodule algebras, we establish a Faddeev-Reshetikhin-Takhtajan (FRT) type presentation, which is written in terms of a K-operator satisfying a reflection algebra relation. The coaction δOq (resp. ιOq) sends the K-operator to a `dressed' K-operator (resp. K-matrix) using Ding-Frenkel type L-operators of LUqsl₂; the counit εOq sends the K-operator to the most general (non-diagonal) solution of the reflection equation. The FRT presentation allows to compute the image via δOq and ιOq of the generating functions of Oq for each presentation (ii)-(iv). Moreover, the images of Lu-Wang's generating functions of Oq in the so-called Drinfeld second realization Dr\! LUqsl₂ are obtained, as well as their images via the (ordinary) coproduct of Dr\! LUqsl₂, which enhance two theorems by Przeździecki. Finally, we apply our FRT presentation to the construction of new type q-characters for LUqsl₂, in imaginary root vectors of Oq, refining those of Frenkel-Reshetikhin, and we explain representation-theoretic meaning of such refining.