The alternating universal K-matrix for the q-Onsager algebra: construct a universal K-matrix for the alternating central extension $A_q$ from solutions of a universal Freidel-Maillet type equation, or prove that none exists

The quantum affine algebra $U_q(\widehat{sl_2})$ admits an 'alternating' tensor product decomposition $U_q(\widehat{sl_2}) \cong \bar{A}_q^{\triangleright} \otimes U_q^{DJ,0} \otimes \bar{A}_q^{\triangleleft}$ in terms of Drinfeld-type 'alternating' subalgebras, derived by Baseilhac from a Freidel-Maillet type presentation of Terwilliger's alternating central extension $\bar{A}_q$ of the positive part $U_q^+$. Since the universal R-matrix of $U_q(\widehat{sl_2})$ is built from the standard triangular decomposition, this raises the question of an 'alternating' universal K-matrix built from a product of solutions to a universal Freidel-Maillet type equation. The natural home for the boundary object is the alternating central extension $A_q$ of the q-Onsager algebra $O_q$, a comodule algebra over the quantum loop algebra. The problem asks for an explicit construction of the universal K-matrix for $A_q$ - an element of a completion of $A_q \otimes \mathcal{L} U_q \mathfrak{sl}_2$ in the sense of the extended Appel-Vlaar framework - satisfying the universal reflection equation with spectral parameter, whose evaluations in the spin-$j$ evaluation representations reproduce the fused K-operators $K^{(j)}(u)$ for all $j \in \frac{1}{2}\mathbb{N}$, and compatible with the coproduct/coaction structure; alternatively, a proof that no such universal object exists. As of 2026 the relation between the fused K-operators and a universal K-matrix for $A_q$ is conjectural, and the universal TT- and TQ-relations derived from the fused K-operators rest on precisely this conjecture.

Background

Quantum affine algebras and their presentations. The quantum affine algebra $U_q(\widehat{sl_2})$ is a Hopf algebra deformation of the universal enveloping algebra of the affine Kac-Moody algebra $\widehat{sl_2}$, central to exactly solvable lattice models and massive integrable quantum field theories. It admits several presentations: the Drinfeld-Jimbo presentation $U_q^{DJ}$ (Chevalley generators $E_i, F_i, K_i^{\pm 1}$, $i = 0, 1$, with quantum Serre relations), the Drinfeld second realization $U_q^{Dr}$ (mode generators $x_k^{\pm}$, Cartan currents, central and derivation generators), the FRT presentation by R-matrix ('RTT') relations, and the equitable presentation (a generating set, introduced by Terwilliger, that treats the positive and negative subalgebras symmetrically instead of triangularly). The positive part $U_q^+$ is the subalgebra generated by $E_0, E_1$; Damiani constructed a Poincare-Birkhoff-Witt (PBW) basis of $U_q^+$ from positive root vectors obtained by Lusztig's braid group action, and the universal R-matrix of $U_q(\widehat{sl_2})$ is built as an ordered product over such root vectors using the tensor product decomposition into positive, Cartan, and negative subalgebras.

Alternating central extensions. Terwilliger described $U_q^+$ by a q-shuffle algebra (an algebra of words in two letters equipped with the shuffle product) and introduced an 'alternating' PBW basis and an alternating central extension $\bar{A}_q$ of $U_q^+$. Baseilhac (2021) obtained a Freidel-Maillet (FM) type presentation of $\bar{A}_q$ and used it to construct the alternating subalgebras $\bar{A}_q^{\triangleright}, \bar{A}_q^{\triangleleft}$ of $U_q^{Dr}$ and the alternating tensor product decomposition $U_q(\widehat{sl_2}) \cong \bar{A}_q^{\triangleright} \otimes U_q^{DJ,0} \otimes \bar{A}_q^{\triangleleft}$, where $U_q^{DJ,0}$ is the Cartan subalgebra. The q-Onsager algebra $O_q$ is the algebra on two generators $W_0, W_1$ obeying the pair of q-Dolan-Grady relations (two cubic q-commutator relations deforming the Dolan-Grady relations that define Onsager's algebra of the two-dimensional Ising model); it appears as the algebra of nonlocal conserved charges of the open XXZ spin chain (the anisotropic spin-1/2 Heisenberg chain with free ends) and of boundary sine-Gordon type models. Its alternating central extension $A_q$ is generated by four families of 'alternating' generators $\{W_{-k}\}$, $\{W_{k+1}\}$, $\{G_{k+1}\}$, $\{\tilde{G}_{k+1}\}$ ($k \ge 0$) and satisfies $A_q \cong O_q \otimes Z$, where $Z$ is a polynomial algebra in countably many central variables (Terwilliger 2021, proving the Baseilhac-Belliard conjecture); Lu-Wang (2021) identified a Drinfeld-type presentation $O_q^{Dr}$ of $O_q$, viewing it as a split $A_1$ iota-quantum group (a quantized enveloping algebra equipped with an involutive automorphism).

Reflection equations, K-matrices, and FM algebras. In integrable systems with boundaries, boundary scattering is encoded by K-matrices: solutions of the spectral-parameter reflection equation, a quadratic relation intertwining a Yang-Baxter solution $R(u)$ with itself. Freidel and Maillet introduced quadratic current algebras whose generating currents satisfy relations of Yang-Baxter/reflection type with operator-valued coefficients; Baseilhac showed that the generating currents of the alternating subalgebras obey FM type relations. A universal K-matrix is a representation-independent ancestor of all K-matrices, the boundary analog of the universal R-matrix. Balagović-Kolb constructed a universal K-matrix for quantum symmetric pairs (coideal subalgebras, i.e. subalgebras $B$ of a Hopf algebra $H$ with $\Delta(B) \subseteq B \otimes H$, invariant under an involution), and Appel-Vlaar extended this to universal coideal subalgebras of quantum Kac-Moody algebras: given a quasitriangular Hopf algebra $H$, an $H$-comodule algebra $B$ (an algebra with a coaction map $B \to B \otimes H$ compatible with the product, the structure dual to a module algebra), and a pair of consistent twists (elements deforming the coproduct in a controlled way), the universal K-matrix is an element of $B \otimes H$ satisfying explicit axioms whose evaluations in finite-dimensional representations solve reflection equations.

Known K-operators for $A_q$. $A_q$ is a comodule algebra over the quantum loop algebra $\mathcal{L} U_q \mathfrak{sl}_2$ (the quotient of $U_q(\widehat{sl_2})$ by the central and degree generators, i.e. the q-deformation of $U(\mathfrak{sl}_2) \otimes \mathbb{C}[z, z^{-1}]$). Lemarthe-Baseilhac-Gainutdinov (2026) developed an extended Appel-Vlaar framework with $H = \mathcal{L} U_q \mathfrak{sl}_2$ and $B = A_q$, proved that for every spin $j \in \frac{1}{2}\mathbb{N}$ there are fused K-operators $K^{(j)}(u)$ satisfying the spectral-parameter reflection equation in the spin-$j$ evaluation representation (the family of representations of the loop algebra labeled by a spectral parameter $u$ and pulled back from the spin-$j$ $U_q(\mathfrak{sl}_2)$-module), and computed them explicitly in terms of the generators of $A_q$ for small $j$. What is missing - the subject of this problem - is the universal object itself, and the proof that the fused K-operators are its evaluations.

Problem Statement

Construct the alternating universal K-matrix for the q-Onsager algebra, or prove that it does not exist.

Setting. Let $\mathcal{L} U_q \mathfrak{sl}_2$ denote the quantum loop algebra in the Drinfeld second realization, and let $A_q$ be the alternating central extension of the q-Onsager algebra $O_q$, with alternating generators $\{W_{-k}\}_{k \ge 0}$, $\{W_{k+1}\}_{k \ge 0}$, $\{G_{k+1}\}_{k \ge 0}$, $\{\tilde{G}_{k+1}\}_{k \ge 0}$, satisfying $A_q \cong O_q \otimes Z$ with $Z$ a polynomial algebra in countably many central variables; $A_q$ is a comodule algebra over $\mathcal{L} U_q \mathfrak{sl}_2$. It is known that: the generating currents of the alternating subalgebras satisfy Freidel-Maillet type quadratic relations (Baseilhac 2021, 2022); and for every $j \in \frac{1}{2}\mathbb{N}$ there exist fused K-operators $K^{(j)}(u)$, expressible in terms of the generators of $A_q$ for small $j$, which satisfy the spectral-parameter reflection equation in the spin-$j$ evaluation representation (Lemarthe-Baseilhac-Gainutdinov 2026).

Positive answer. Construct an element $\mathcal{K}(u)$ of a completion of $A_q \otimes \mathcal{L} U_q \mathfrak{sl}_2$ (in the sense of the extended Appel-Vlaar axiomatics, i.e. as a universal K-matrix associated with the comodule algebra $B = A_q$ over $H = \mathcal{L} U_q \mathfrak{sl}_2$ and a pair of consistent twists), such that:

(i) $\mathcal{K}(u)$ satisfies the universal reflection equation with spectral parameter, compatible with the Freidel-Maillet type relations satisfied by the alternating currents - in the spirit of Baseilhac's proposal [1], built as a product of solutions of a universal Freidel-Maillet type equation, the boundary analog of the construction of the universal R-matrix from root vectors;

(ii) for every $j \in \frac{1}{2}\mathbb{N}$, the evaluation of $\mathcal{K}(u)$ in the spin-$j$ evaluation representation coincides, up to a scalar normalization, with the fused K-operator $K^{(j)}(u)$;

(iii) $\mathcal{K}(u)$ is compatible with the coalgebra structure: it intertwines the coaction of $A_q$ and satisfies the expected tensor-product relations under the coproduct of $\mathcal{L} U_q \mathfrak{sl}_2$ (for instance Damiani's topological Drinfeld coproduct, or the corresponding limit of the Drinfeld-Jimbo coproduct), so that the universal TT- and TQ-relations for $A_q$ (the functional relations among the spin-$j$ transfer matrices constructed from the $K^{(j)}(u)$) follow without the technical conjecture currently used in the literature.

Negative answer. Prove instead that no element of the natural completion satisfies (i)-(ii) simultaneously, i.e. that the fused K-operators of different spins cannot all arise as evaluations of a single universal K-matrix for $A_q$.

A proof of the evaluation relation (ii) alone, with the universal reflection equation (i) established, already constitutes a substantial resolution: it would remove the conjecture on which the published universal TT- and TQ-relations currently rest.

References

1. P. Baseilhac, 'The alternating presentation of $U_q(\widehat{gl_2})$ from Freidel-Maillet algebras', Nucl. Phys. B 967 (2021) 115400. arXiv:2011.01572; doi:10.1016/j.nuclphysb.2021.115400; https://doi.org/10.1016/j.nuclphysb.2021.115400. (Poses the alternating universal K-matrix problem in its introduction.)
2. P. Baseilhac, 'Freidel-Maillet type presentations of $U_q(sl_2)$', Nucl. Phys. B 975 (2022) 115682. arXiv:2106.11715; doi:10.1016/j.nuclphysb.2022.115682; https://doi.org/10.1016/j.nuclphysb.2022.115682. (States that finding the universal K-matrix for the equitable presentation is an open problem.)
3. P. Terwilliger, 'The alternating central extension of the q-Onsager algebra', Commun. Math. Phys. 387 (2021) 1771-1819. arXiv:2103.03028; doi:10.1007/s00220-021-04171-2; https://doi.org/10.1007/s00220-021-04171-2. (Proves the Baseilhac-Belliard conjecture $A_q \cong O_q \otimes Z$.)
4. P. Terwilliger, 'The q-Onsager algebra and its alternating central extension', Nucl. Phys. B 975 (2022) 115662. arXiv:2106.14041; doi:10.1016/j.nuclphysb.2022.115662; https://doi.org/10.1016/j.nuclphysb.2022.115662.
5. P. Terwilliger, 'The alternating central extension for the positive part of $U_q(\widehat{sl_2})$', Nucl. Phys. B 947 (2019) 114729. arXiv:1907.09872; doi:10.1016/j.nuclphysb.2019.114729; https://doi.org/10.1016/j.nuclphysb.2019.114729.
6. M. Lu, W. Wang, 'A Drinfeld type presentation of affine iota-quantum groups I: split ADE type', Adv. Math. 393 (2021) 108111. arXiv:2009.04542; doi:10.1016/j.aim.2021.108111; https://doi.org/10.1016/j.aim.2021.108111.
7. G. Lemarthe, P. Baseilhac, A. M. Gainutdinov, 'Fused K-operators and the q-Onsager algebra', SIGMA 22 (2026) 026. arXiv:2301.00781; doi:10.3842/SIGMA.2026.026; https://doi.org/10.3842/SIGMA.2026.026. (Extended Appel-Vlaar framework with $H = \mathcal{L} U_q \mathfrak{sl}_2$, $B = A_q$; fused K-operators; evaluation relation stated as conjecture.)
8. P. Baseilhac, A. M. Gainutdinov, G. Lemarthe, 'Universal TT- and TQ-relations via centrally extended q-Onsager algebra', Nucl. Phys. B 1026 (2026) 117427. arXiv:2511.15876; doi:10.1016/j.nuclphysb.2026.117427; https://doi.org/10.1016/j.nuclphysb.2026.117427. (Universal TT-/TQ-relations derived within a technical conjecture.)
9. A. Appel, B. Vlaar, 'Universal K-matrices for quantum Kac-Moody algebras', Represent. Theory 26 (2022) 764-824. arXiv:2007.09218; doi:10.1090/ert/623; https://doi.org/10.1090/ert/623.
10. M. Balagović, S. Kolb, 'Universal K-matrix for quantum symmetric pairs', J. reine angew. Math. 747 (2019) 299-353. arXiv:1507.06276; doi:10.1515/crelle-2016-0012; https://doi.org/10.1515/crelle-2016-0012.
11. I. Damiani, 'On the Drinfeld coproduct', Pure Appl. Math. Q. 20 (2024), no. 1, 171-232. doi:10.4310/PAMQ.2024.v20.n1.a6; https://doi.org/10.4310/PAMQ.2024.v20.n1.a6. (Topological Drinfeld coproduct for general quantum affinizations.)
12. I. Damiani, 'From the Drinfeld realization to the Drinfeld-Jimbo presentation of affine quantum algebras: injectivity', Publ. Res. Inst. Math. Sci. 51 (2015) 131-171. arXiv:1407.0341; doi:10.4171/prims/150; https://doi.org/10.4171/prims/150.
13. L. Freidel, J.-M. Maillet, 'Quadratic algebras and integrable systems', Phys. Lett. B 262 (1991) 278-284. doi:10.1016/0370-2693(91)91566-E; https://doi.org/10.1016/0370-2693(91)91566-E.
14. V. Chari, A. Pressley, 'Quantum affine algebras', Commun. Math. Phys. 142 (1991) 261-283. doi:10.1007/bf02102063; https://doi.org/10.1007/bf02102063. (Section 4.4: the partial information on the coproduct in the Drinfeld presentation.)