Freidel–Maillet type presentations beyond the positive part of $U_q(\widehat{sl_2})$: the equitable-type presentation of the full quantum affine algebra, higher-rank extensions, and the complete Hopf isomorphism between its Drinfeld–Jimbo and Drinfeld realizations

The quantum affine algebra $U_q(\widehat{sl_2})$ admits a Drinfeld–Jimbo (Chevalley) presentation, Drinfeld's second (current) realization, an FRT/R-matrix presentation, and an equitable presentation. Baseilhac (2022) proved that the nonstandard positive part of $U_q(\widehat{sl_2})$ of equitable type admits a Freidel–Maillet type presentation — a K-operator satisfying a generalized reflection equation together with a quantum determinant condition — and embedded it into Drinfeld's second realization. The paper's concluding remarks formulate three open directions, and its appendix flags the classical problem of a complete Hopf algebra isomorphism between the Drinfeld–Jimbo and Drinfeld realizations. All four remain open as of August 2026: (i) the Freidel–Maillet presentation of the full algebra $U_q(\widehat{sl_2})$ of equitable type, announced for future work; (ii) Freidel–Maillet and alternating presentations for higher-rank affine Lie algebras; (iii) quantum integrable models built from Freidel–Maillet type presentations; and (iv) the complete Hopf algebra isomorphism problem flagged in the paper's appendix. None of the four components is resolved: the closest follow-up (Ruan 2025) remains at the level of the positive part, and the coproduct question is the object of active but explicitly partial work (Beck 1994; Khoroshkin–Tolstoy 1994; Damiani 1998; Finkelberg–Tsymbaliuk 2019; Zhang 2024; Milot 2025, 2026; Neguţ 2026).

Background

The quantum affine algebra $U_q(\widehat{sl_2})$ — the quantum group (a one-parameter deformation of the universal enveloping algebra of a Lie algebra) associated with the affine Kac–Moody algebra $\widehat{sl_2}$ (the central extension of the Lie algebra of $sl_2$-valued loops on a circle) and the algebraic backbone of two-dimensional quantum integrable models with $sl_2$-type symmetry — admits several equivalent but structurally different presentations. In the Drinfeld–Jimbo presentation $U_q^{DJ}$ it is generated by Chevalley elements $E_i,F_i,K_i^{\pm1}$ ($i=0,1$) subject to $q$-Serre relations (the $q$-deformed analogues of the Serre relations defining semisimple Lie algebras). In Drinfeld's second realization $U_q^{Dr}$ it is generated by currents $x^\pm(z)=\sum_k x^\pm_k z^{-k}$ and a Cartan current $\theta(z)$, with relations written for generating series; this is the form in which highest-weight modules, $q$-vertex operators (intertwining operators between current-algebra modules) and Bethe-ansatz symmetry algebras are most naturally described; the Bethe ansatz is the standard method for diagonalizing integrable-model transfer matrices, parametrizing eigenvectors through rapidity variables. The two presentations are known to be isomorphic as algebras. A Hopf algebra is a bialgebra with an antipode; its coproduct $\Delta$ is an algebra homomorphism encoding how tensor products of representations are formed. What has no known closed form is the behaviour of the coproduct under the isomorphism $U_q^{DJ}\simeq U_q^{Dr}$: the Drinfeld–Jimbo coproduct, transported to the current generators, admits no known closed formula in the currents, and such formulas are only meaningful in completed tensor products (topological extensions of the algebraic tensor product in which infinite current expansions converge as formal series).

A third family of presentations comes from integrable systems. The FRT (Faddeev–Reshetikhin–Takhtajan) or $RLL$ presentation realizes $U_q(\widehat{gl_2})$ through an L-operator — a $2\times2$ matrix whose entries are generators — obeying the Yang–Baxter relation (an $RLL=LLR$-type quadratic compatibility condition) with the trigonometric R-matrix $R(u)$ (the matrix solution of the Yang–Baxter equation that encodes the integrability of the model); Ding and Frenkel established the isomorphism between the $RLL$ and Drinfeld realizations in type A. Freidel and Maillet introduced quadratic algebras, now called Freidel–Maillet (FM) algebras, generated by the entries of a K-operator $K(u)$ satisfying a generalized reflection equation (the boundary analogue of the Yang–Baxter equation) involving both the spectral-parameter R-matrix $R(u/v)$ and a constant R-matrix $R^{(0)}$, together with a quantum determinant condition (the $q$-analogue of the determinant of a matrix of generators, fixed to a scalar or a prescribed series). These algebras control boundary degrees of freedom and coactions in integrable models with boundaries: a coaction of a Hopf algebra $H$ on an algebra $A$ is an algebra map $A\to A\otimes H$ compatible with the coproduct, encoding a family of endomorphisms of $A$ parametrized by $H$.

The equitable presentation, introduced by Ito and Terwilliger in the study of tridiagonal pairs (pairs of diagonalizable linear maps on a finite-dimensional vector space that act on each other's eigenspaces in a tridiagonal fashion), presents $U_q(\widehat{sl_2})$ and its positive part (the subalgebra generated by the positive current modes) through generators placed in symmetric roles, in contrast with the asymmetric $E,F,K$ of Chevalley type. Terwilliger constructed the alternating central extension $\mathcal U_q^+$ of the positive part $U_q^+$, whose alternating elements (sign-alternating sums over the current indices) generate a subalgebra closely related to the $q$-Onsager algebra, the symmetry algebra of the spin-$1/2$ XXZ chain (the anisotropic one-dimensional Heisenberg quantum spin chain) with generic boundary conditions.

Baseilhac (2022) proved that the nonstandard subalgebra $U_q^{T,+}$ of $U_q^+$ generated by the alternating elements of the equitable currents admits a Freidel–Maillet type presentation: a K-operator whose entries are generating functions of those currents, satisfying the FM equation and a quantum determinant condition, with an explicit embedding into Drinfeld's second realization and a coaction of the Drinfeld–Jimbo-type FM algebra. For the Drinfeld–Jimbo type, an FM presentation of (the alternating subalgebra of) $U_q(\widehat{gl_2})$ was previously obtained by the same author from the FRT presentation and a Drinfeld twist (an invertible element modifying the coproduct while preserving the algebra), and for the finite algebra $U_q(sl_2)$ the FM presentation unifying the Drinfeld–Jimbo and equitable types is known from the companion paper. The equitable affine full-algebra case is the announced missing piece, and the higher-rank, integrable-model and Hopf-level questions complete the open landscape surveyed here.

Problem Statement

The concluding remarks of the source paper announce as future work the extension of the Freidel–Maillet presentation framework to the full algebra $U_q(\widehat{sl_2})$ of equitable type, and pose further expected extensions by analogy; the paper's appendix records a standalone classical structural question about the same two standard presentations. Together they define the following four open questions. All four are mandatory and independent: resolving any one of them does not resolve the others, and each is resolved only by satisfying its own completion condition stated below.

(1) Construct the Freidel–Maillet type presentation of the full quantum affine algebra $U_q(\widehat{sl_2})$ of equitable type. Concretely: find an equitable-type K-operator — a $2\times2$ matrix whose entries are generating functions in the equitable generators of the full algebra — satisfying the Freidel–Maillet equation (the generalized reflection equation involving both the spectral-parameter R-matrix and the constant R-matrix) together with a quantum determinant condition, such that the algebra defined by these data is isomorphic to $U_q(\widehat{sl_2})$ in its equitable presentation; characterize the resulting embedding into Drinfeld's second realization and the associated coaction structure. Constructing instead the analogous equitable-type Freidel–Maillet presentation of the central extension $U_q(\widehat{gl_2})$, with its embedding into Drinfeld's second realization, is an acceptable equivalent that independently resolves this component. The Drinfeld–Jimbo-type analogue is known (via the FRT presentation and a Drinfeld twist) and the finite $U_q(sl_2)$ case is known; the affine equitable case is the announced missing piece.

(2) Generalize the Freidel–Maillet and alternating presentations (of Drinfeld–Jimbo or equitable type) to higher-rank affine Lie algebras, in analogy with the recently achieved $RLL$-to-Drinfeld isomorphisms in types B, C, D and twisted cases. Concretely: construct, for at least one affine type of rank greater than one, the Freidel–Maillet presentation — defined by its $R$-matrices, $K$-operator and quantum determinant condition — together with the corresponding alternating presentation, and verify that the algebra defined in each case is isomorphic to the corresponding quantum affine algebra.

(3) Construct and study quantum integrable models whose defining algebraic structure is a Freidel–Maillet type presentation — for example boundary spin chains, or models solvable by Bethe-ansatz or $q$-vertex-operator techniques built from an FM algebra. Concretely: exhibit at least one such model, identifying its transfer matrix and the Bethe-ansatz or $q$-vertex-operator data that make it solvable. The source notes that FRT presentations are known to provide a powerful framework for the explicit construction and analysis of quantum integrable models such as spin chains, using for instance Bethe ansatz or $q$-vertex-operator techniques, and that it would be natural to study the analogous models built from Freidel–Maillet type presentations.

(4) Settle the long-standing structural problem noted in the source's appendix: construct the complete Hopf algebra isomorphism between the Drinfeld–Jimbo presentation $U_q^{DJ}$ and Drinfeld's second realization $U_q^{Dr}$ of $U_q(\widehat{sl_2})$ — an identification at the level of the coproducts, with closed coproduct formulas for the current generators in an appropriate completion of the tensor product (a topological extension in which the infinite sums defining the transported coproduct converge formally), beyond the known algebra-level isomorphisms and the partial coproduct information available in the literature.

The source poses these items in different registers: component (1) is the sharply posed construction problem that the paper announces for future work ("Details will be considered elsewhere"); components (2) and (3) are expected extensions and research directions posed by analogy; and component (4) is a standalone classical open problem recorded in the paper's appendix immediately after the recalled algebra-level isomorphism $U_q^{DJ}\to U_q^{Dr}$, concerning the same two standard presentations that the framework's embeddings and comparisons use throughout.

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