Universal K-matrix for the alternating central extension of the $q$-Onsager algebra: existence and identification with fused K-operators (Conjecture 1 of Lemarthe--Baseilhac--Gainutdinov)

The alternating central extension $\mathcal{A}_q$ of the $q$-Onsager algebra is a right comodule algebra over the quantum loop algebra $\mathcal{L}U_q\mathfrak{sl}_2$. Lemarthe, Baseilhac and Gainutdinov axiomatized a universal K-matrix $\mathfrak{K}$ for such comodule algebras -- an element of an appropriate completion of $\mathcal{A}_q\otimes\mathcal{L}U_q\mathfrak{sl}_2$ satisfying three axioms (K1)-(K3) involving a twist automorphism -- and, without assuming its existence, constructed fused K-operators $\mathcal{K}^{(j)}(u)$ of arbitrary spin $j$ that provably satisfy the spectral-parameter reflection equation. Their Conjecture 1 states that spin-$j$ evaluations of the (conjectural) universal K-matrix reproduce the fused K-operators up to an explicit central series $\nu(u)$ determined by a functional relation with the quantum determinant, and that $\nu(u)$ carries a specific evaluated coaction. The open problem is to construct a universal K-matrix for $\mathcal{A}_q$ with the twist used in that work (or prove that none exists) and to prove Conjecture 1, including the universal-level coaction of $\nu(u)$. The problem remains open.

Background

Integrable quantum systems with boundaries are governed by the reflection equation. Given a solution $R(u)$ of the Yang--Baxter equation on an auxiliary space $V$ with spectral parameter $u$, a K-matrix $K(u)\in\mathrm{End}(V)$ solving $R(u/v)K_1(u)R(uv)K_2(v)=K_2(v)R(uv)K_1(u)R(u/v)$ supplies integrable boundary conditions for open spin chains: by Sklyanin's construction, transfer matrices built from $R$ and $K$ mutually commute and generate the conserved quantities. When the entries of $K$ lie in a noncommutative algebra rather than in the ground field one speaks of a K-operator; K-operators are the representation-independent origin of K-matrices, because further evaluating their entries in a representation produces concrete K-matrices.

The $q$-Onsager algebra $O_q$ is a $q$-deformation of Onsager's Lie algebra (the symmetry algebra of the two-dimensional Ising model); it arises as a coideal subalgebra of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_2)$ (a subalgebra stable under the coproduct, $\Delta(B)\subset B\otimes U_q(\widehat{\mathfrak{sl}}_2)$) and is the hidden symmetry of the open XXZ spin chain with generic integrable boundary conditions (Baseilhac--Koizumi). Its alternating central extension $\mathcal{A}_q$ adjoins the 'imaginary' alternating generators so that the enlarged algebra becomes a right comodule algebra over the quantum loop algebra $H=\mathcal{L}U_q\mathfrak{sl}_2$ (the version of the quantum affine algebra without the central element and derivation): there is a coaction $\delta:\mathcal{A}_q\to\mathcal{A}_q\otimes H$ compatible with the product. A key structural fact is that $\mathcal{A}_q$ admits a presentation as a reflection algebra (Baseilhac--Shigechi, refined in the source work): the $2\times2$ matrix $\mathcal{K}^{(1/2)}(u)$ whose entries are the generating functions $\mathcal{W}_\pm(u)$, $\mathcal{G}_\pm(u)$ of the alternating generators, together with two boundary parameters $k_\pm$, satisfies the reflection equation, and these matrix relations are a complete set of defining relations for $\mathcal{A}_q$.

For a quasitriangular Hopf algebra $H$ with universal R-matrix $\mathfrak{R}$, evaluating $\mathfrak{R}$ on tensor products of the evaluation representations $\pi^j_u$ of spin $j$ (the finite-dimensional $\mathcal{L}U_q\mathfrak{sl}_2$-modules of dimension $2j+1$, $j\in\frac12\mathbb{N}$) produces all spin-$j$ R-matrices up to an explicit scalar central series $\mu(u)$. The analogous boundary object is the universal K-matrix. Kolb and Appel--Vlaar constructed such matrices for quantum symmetric pairs, where the boundary algebra is a coideal subalgebra of $H$; Lemarthe--Baseilhac--Gainutdinov (following a suggestion going back to Enriquez's reflection bialgebras) extended the axioms to comodule algebras $B$: given a twist pair $(\psi,J)$ -- an algebra automorphism $\psi$ of $H$ and a Drinfeld twist $J\in H\otimes H$ satisfying a compatibility condition with $\mathfrak{R}$ -- a universal K-matrix is an element $\mathfrak{K}$ of an appropriate completion of $B\otimes H$ obeying, for all $b\in B$, $\mathfrak{K}\,\delta(b)=\delta^\psi(b)\,\mathfrak{K},\qquad \delta^\psi=(\mathrm{id}\otimes\psi)\circ\delta \ \ (\mathrm{K1}),$ $(\delta\otimes\mathrm{id})(\mathfrak{K})=(\mathfrak{R}^\psi)_{32}\,\mathfrak{K}_{13}\,\mathfrak{R}_{23},\qquad \mathfrak{R}^\psi=(\psi\otimes\mathrm{id})(\mathfrak{R}) \ \ (\mathrm{K2}),$ $(\mathrm{id}\otimes\Delta)(\mathfrak{K})=J_{23}^{-1}\,\mathfrak{K}_{13}\,\mathfrak{R}_{23}^{\psi}\,\mathfrak{K}_{12} \ \ (\mathrm{K3}),$ where $\Delta$ is the coproduct and indices denote tensor legs. Axiom (K1) is a twisted intertwining relation between the coaction and its $\psi$-twist, (K2) constrains the coaction of $\mathfrak{K}$, and (K3) its coproduct behavior; on representations they imply the reflection equation. The axioms specialize to those of Kolb and of Appel--Vlaar in the coideal case.

For $B=\mathcal{A}_q$, $H=\mathcal{L}U_q\mathfrak{sl}_2$ and the twist pair $(\psi,J)=(\eta,1\otimes1)$, where $\eta$ is the involutive automorphism exchanging the two Chevalley generators ($\eta(E_0)=F_1$, $\eta(E_1)=F_0$, $\eta(K_0^{1/2})=K_1^{-1/2}$, $\eta(K_1^{1/2})=K_0^{-1/2}$), the existence of $\mathfrak{K}$ is conjectural; this choice of twist is made so that the evaluated equations reproduce the standard (principal-gradation) reflection equation, and it lies outside the twist families covered by the existence theorems of Appel--Vlaar. Independently of existence, the same authors construct fused K-operators $\mathcal{K}^{(j)}(u)\in\mathcal{A}_q[[u^{-1}]]\otimes\mathrm{End}(\mathbb{C}^{2j+1})$ for all $j\in\frac12\mathbb{N}$ by a fusion recursion built from the fundamental K-operator $\mathcal{K}^{(1/2)}(u)$, the fused R-matrices and explicit intertwiners $\mathcal{E}^{(j+1/2)}$, $\mathcal{F}^{(j+1/2)}$ of the tensor-product evaluation representations, and they prove that the fused K-operators satisfy the reflection equations for any pair of spins, with no assumption on $\mathfrak{K}$. Conjecture 1 then relates the two pictures: assuming $\mathfrak{K}$ exists with spin-$j$ evaluations $\mathbf{K}^{(j)}(u)=(\mathrm{id}\otimes\pi^j_{u^{-1}})(\mathfrak{K})$, it asserts $\mathbf{K}^{(j)}(u)=\nu^{(j)}(u)\,\mathcal{K}^{(j)}(u)\qquad\text{for all } j\in\tfrac12\mathbb{N},$ where $\nu(u)\equiv\nu^{(1/2)}(u)$ is an invertible central element of $\mathcal{A}_q[[u^{-1}]]$ defined by the functional relation $\pi^{1/2}(\mu(u^2q))\,\nu(u)\nu(uq)\Gamma(u)=1$ with $\Gamma(u)$ the quantum determinant of the fundamental K-operator (a generating function of central elements of $\mathcal{A}_q$), where $\nu^{(j)}(u)$ is the explicit fused product of $\nu$ and $\mu$-factors given in the conjecture, and where $\nu(u)$ carries the evaluated coaction $\delta_w(\nu(u))=(u^2q+u^{-2}q^{-1})\nu(u)\otimes\mu(u/w)\mu(uw)$. The supporting evidence is substantial: the modes of $\nu(u)$ are proven to be uniquely determined (up to the sign of the constant term) by a recursion in the central generators, and the normalized products $\tilde{\mathbf{K}}^{(j)}(u)=\nu^{(j)}(u)\mathcal{K}^{(j)}(u)$ are proven to satisfy the evaluated forms of all three axioms (K1')-(K3') and the reflection equations at every spin. Standing assumptions: $q\in\mathbb{C}^*$ is not a root of unity.

Problem Statement

Construct a universal K-matrix for the comodule algebra $\mathcal{A}_q$ (the alternating central extension of the $q$-Onsager algebra) over the quantum loop algebra $H=\mathcal{L}U_q\mathfrak{sl}_2$, in the sense of Lemarthe--Baseilhac--Gainutdinov: exhibit an element $\mathfrak{K}$ of an appropriate completion of $\mathcal{A}_q\otimes H$ satisfying the axioms (K1)-(K3) for the twist pair $(\psi,J)=(\eta,1\otimes1)$ used in that work, where $\eta$ is the involutive automorphism of $\mathcal{L}U_q\mathfrak{sl}_2$ given by $\eta(E_0)=F_1$, $\eta(E_1)=F_0$, $\eta(K_0^{1/2})=K_1^{-1/2}$, $\eta(K_1^{1/2})=K_0^{-1/2}$; a proof that no such element exists in any reasonable completion would equally resolve the fundamental question; a full answer to the existence question therefore requires a genuinely nondegenerate object, so we require $\mathfrak{K}\neq 0$ and its spin-$j$ reductions $(\mathrm{id}\otimes\pi^j_{u^{-1}})(\mathfrak{K})$ to be nonzero for all $j\in\frac12\mathbb{N}$, excluding the trivial solution $\mathfrak{K}=0$, which satisfies every relation trivially because all terms in (K1)-(K3) vanish at the zero element; note that (K3) is quadratic in $\mathfrak{K}$ (unlike the linear (K1) and (K2)), so the solution set is not a linear subspace and this degenerate case is not removed by a scalar normalization. In the affirmative case, prove Conjecture 1 of that paper: for every $j\in\frac12\mathbb{N}$, the spin-$j$ evaluation $\mathbf{K}^{(j)}(u)=(\mathrm{id}\otimes\pi^j_{u^{-1}})(\mathfrak{K})$ equals $\nu^{(j)}(u)\mathcal{K}^{(j)}(u)$, where $\mathcal{K}^{(j)}(u)$ is the fused K-operator defined by the fusion recursion from the fundamental K-operator, $\nu(u)$ is the invertible central element of $\mathcal{A}_q[[u^{-1}]]$ determined by the functional relation $\pi^{1/2}(\mu(u^2q))\nu(u)\nu(uq)\Gamma(u)=1$ (equivalently, by its proven recursive mode formulas), and $\nu^{(j)}(u)$ is the explicit fused product of $\nu$ and $\mu$-factors given in the conjecture; by the reduction of Baseilhac--Gainutdinov--Lemarthe it suffices to establish the identification for $j=1/2$, the general case then following from the fusion/reduction relations. As an integral part of the identification, derive the coaction of $\nu(u)$ at the universal level so that it reproduces the evaluated coaction $\delta_w(\nu(u))=(u^2q+u^{-2}q^{-1})\nu(u)\otimes\mu(u/w)\mu(uw)$. The problem is posed at the natural generality of the source: $q\in\mathbb{C}^*$ not a root of unity, arbitrary central character of $\mathcal{A}_q$, all spins $j\in\frac12\mathbb{N}$, and no restriction of the completion beyond what the axioms require.

References

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