{"arxiv_id":"2611.12874","title":"FRT presentation of the $q$-Onsager comodule algebra and refined $q$-characters","abstract":"We introduce a concept of counital comodule algebra $(B,\\delta_B,\\epsilon_B)$ over a Hopf ${\\mathbb K}$-algebra $H$, where $\\delta_B: B\\rightarrow B \\otimes H$ and $\\epsilon_B: B\\rightarrow {\\mathbb K}$ are coaction and counit homomorphisms, respectively. This structure gives a canonical (counital) comodule algebra map $\\iota_B: B \\rightarrow H$. In this paper, we study in details the case $B=O_q$, and $H$ is the quantum loop algebra ${\\cal L}U_qsl_2$ for ${\\mathbb K}={\\mathbb C}$. Here, $O_q$ 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 $(O_q,\\delta_{O_q},\\epsilon_{O_q})$, 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 $\\delta_{O_q}$ (resp. $\\iota_{O_q}$) sends the K-operator to a `dressed' K-operator (resp. K-matrix) using Ding-Frenkel type L-operators of ${\\cal L}U_qsl_2$; the counit $\\epsilon_{O_q}$ sends the K-operator to the most general (non-diagonal) solution of the reflection equation. The FRT presentation allows to compute the image via $\\delta_{O_q}$ and $\\iota_{O_q}$ of the generating functions of $O_q$ for each presentation (ii)-(iv). Moreover, the images of Lu-Wang's generating functions of $O_q$ in the so-called Drinfeld second realization ${}^{Dr}\\!{\\cal L}U_qsl_2$ are obtained, as well as their images via the (ordinary) coproduct of ${}^{Dr}\\!{\\cal L}U_qsl_2$, which enhance two theorems by Prze\\'zdziecki. Finally, we apply our FRT presentation to the construction of new type $q$-characters for ${\\cal L}U_qsl_2$, in imaginary root vectors of $O_q$, refining those of Frenkel-Reshetikhin, and we explain representation-theoretic meaning of such refining.","primary_cat":"math.QA","cross_cats":["math-ph","math.MP","math.RT"],"submitted_at":"2026-11-09T18:57:36+00:00","latest_version":1,"source":"arxiv","source_label":"arXiv","source_url":null,"has_pdf":true,"has_source":false,"verdict":"CONDITIONAL","confidence":"MODERATE","one_line_summary":"The q-Onsager algebra admits an FRT type presentation as a counital comodule algebra over the quantum loop algebra, leading to refined q-characters that expose Grothendieck ring relations.","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","upvotes":0,"downvotes":0,"pith_score":0,"inbound_count":0,"_links":{"html":"https://pith.science/paper/2611.12874","verdict":"/api/v1/verdicts/2611.12874"}}