A Lusztig-type quantum group structure for the q-Onsager algebra at roots of unity: presentation, cyclic tridiagonal pairs, and non-abelian symmetry sectors of the open XXZ chain

The $q$-Onsager algebra $\mathcal O_q$ is the two-generator, two-relation $q$-deformation of the Onsager algebra that governs Onsager-type non-abelian symmetry in boundary quantum integrable models such as the open XXZ spin chain; for generic $q$ its representation theory is organized by tridiagonal pairs and Leonard (Askey-scheme) duality. Vu's 2016 thesis posed the root-of-unity counterpart of Lusztig's quantum group construction: at $q=e^{i\pi/N}$ the generators $W_0, W_1$ together with their divided polynomials $W_0^{[N]}, W_1^{[N]}$ generate a new algebra, conjecturally presented by four families of relations, that should play the role of the small quantum group enlarged by Lusztig divided powers; its structure (Hopf-type or coideal), its finite-dimensional irreducible representations — connected with the newly proposed notion of cyclic tridiagonal pair — and the induced non-abelian symmetry sectors of the open XXZ chain at roots of unity were all left open. The companion conjecture on higher-order $q$-Dolan-Grady relations has been proved for $\mathcal O_q$ by Terwilliger (2018, via the Baseilhac-Kolb braid group action) and subsumed for general quantum symmetric pairs by the Serre-Lusztig theorems of Chen-Letzter-Lu-Wang (2021-2023); the cyclic tridiagonal pair framework was introduced with a still-conjectural presentation by Baseilhac-Gainutdinov-Vu (2017). The open problem addressed here is: establish the presentation and structure of this root-of-unity algebra, classify its irreducible modules and, as a related problem, the irreducible cyclic tridiagonal pairs, and use the classification to characterize the non-abelian symmetry sectors of the open XXZ chain at roots of unity.

Background

The Onsager algebra is the infinite-dimensional Lie algebra with two generators $A, A^*$ subject to the trilinear Dolan-Grady relations $[A,[A,[A,A^*]]]=16[A,A^*]$ and $[A^*,[A^*,[A^*,A]]]=16[A^*,A]$, where $[X,Y]=XY-YX$; it first appeared in the exact solution of the two-dimensional Ising model, and Dolan and Grady showed that these two relations alone generate a commuting family of conserved charges — the integrable hierarchy behind the Ising and related models. The $q$-Onsager algebra $\mathcal O_q$ is its $q$-deformation: the associative algebra with two generators $W_0, W_1$ satisfying the cubic $q$-Dolan-Grady relations $[W_i,[W_i,[W_i,W_{i+1}]_q]_{q^{-1}}]=\rho_i[W_i,W_{i+1}]$ for $i=0,1$ (indices modulo 2), where $[X,Y]_q=qXY-q^{-1}YX$; at $q=1$, $\rho_i=16$ these reduce to the Dolan-Grady relations. $\mathcal O_q$ is the non-abelian symmetry algebra of boundary integrable models, notably the open spin-1/2 XXZ chain with generic integrable boundary conditions (boundary integrability being governed by the reflection equation, the boundary analogue of the Yang-Baxter equation that supplements it at the chain's ends), whose mutually commuting transfer matrices and local conserved quantities (including the Hamiltonian) are polynomials in two elements of $\mathcal O_q$ (Baseilhac-Koizumi 2005), so that spectral problems reduce to the representation theory of $\mathcal O_q$.

For generic $q$, that representation theory is organized by tridiagonal pairs: ordered pairs $(A,A^*)$ of diagonalizable linear maps on a finite-dimensional vector space such that each eigenspace of one map is carried by the other only into the sum of the corresponding eigenspace and its two neighbors in a natural ordering, and the two have no common invariant subspace. When all eigenspaces are one-dimensional this is a Leonard pair, and the change-of-basis coefficients between the two eigenbases are orthogonal polynomials of the Askey scheme ($q$-Racah and degeneracies) — Leonard duality. The finite-dimensional irreducible $\mathcal O_q$-modules correspond to tridiagonal pairs of $q$-Racah type.

When $q$ is a root of unity, $q=e^{i\pi/N}$, ordinary quantum groups degenerate in a controlled way: Lusztig introduced divided powers $E^{(n)}=E^n/[n]_q!$, with $[n]_q=(q^n-q^{-n})/(q-q^{-1})$, and constructed the finite-dimensional small quantum group and the enlarged full quantum group, whose non-semisimple representation theory underlies the special behavior of quantum models at roots of unity. No analogue of this construction existed for $\mathcal O_q$. Vu's thesis introduced divided polynomials $W_i^{[N]}$ — suitably normalized degree-$N$ polynomials in the generators $W_i$ — which remain well defined at $q=e^{i\pi/N}$ and, on representations arising from the XXZ chain, encode a cyclic degeneration of the tridiagonal structure: the ordered eigenspaces wrap around, $V_{-1}=V_{d-1}$ and $V_0=V_d$ (diameter $d$), instead of terminating. This motivated the new notion of a cyclic tridiagonal pair — the tridiagonal-pair axioms with the neighbor ordering taken cyclically (thesis Definition 4.2.1; in Baseilhac-Gainutdinov-Vu 2017, Definition 1.1, the eigenspaces are indexed by the cyclic group $\mathbb Z_N$, with wrap-around $V_{-1}=V_{N-1}$, $V_0=V_N$). The four operators $W_0, W_1, W_0^{[N]}, W_1^{[N]}$ generate a new algebra whose relations resemble the higher-order $q$-Dolan-Grady relations that Baseilhac and Vu conjectured for $\mathcal O_q$ and for its ADE-type generalizations (generalized $q$-Onsager algebras associated with simply-laced affine Dynkin diagrams, identifiable with the ı quantum groups, i.e. the coideal subalgebras attached to quantum symmetric pairs of Drinfeld-Jimbo quantum groups).

Parts of this agenda have since been settled. Terwilliger proved the Baseilhac-Vu conjecture on higher-order $q$-Dolan-Grady relations for $\mathcal O_q$, using the braid group action and root vectors constructed by Baseilhac and Kolb; the Chen-Letzter-Lu-Wang Serre-Lusztig theorems establish the analogous higher-order relations for ı quantum groups of arbitrary Kac-Moody type, which covers the ADE and non-simply-laced generalizations in a different formulation. Baseilhac-Gainutdinov-Vu developed the cyclic tridiagonal pair framework: explicit examples, the identification of the algebra generated by the divided polynomials with a higher-order generalization of the Onsager algebra, and a conjectural four-family presentation of the specialization of $\mathcal O_q$ at $q=e^{i\pi/N}$ (the $q$-Serre-type relations, degree-$N$ polynomial relations, mixed relations, and higher-order Onsager relations). What remains open is the root-of-unity structure theory itself: a rigorous presentation, the Hopf-type or coideal structure, the classification of finite-dimensional irreducible modules and, as a related problem, of irreducible cyclic tridiagonal pairs — and the consequent characterization of the non-abelian symmetry sectors of the open XXZ chain at roots of unity.

Problem Statement

Construct the analog of Lusztig's quantum group framework for the q-Onsager algebra at roots of unity, as posed in Vu's thesis (Perspectives 4.2 and 4.3 and the unfinished Chapter 3 project). Let $\mathcal O_q$ be the $q$-Onsager algebra with generators $W_0, W_1$ and $q$-Dolan-Grady relations, let $q=e^{i\pi/N}$ with integer $N\ge 2$, and let $W_0^{[N]}, W_1^{[N]}$ denote the divided polynomials of the two fundamental generators (as defined in the Background: suitably normalized degree-$N$ polynomials in $W_0$ and $W_1$ that remain well defined at $q=e^{i\pi/N}$, the $\mathcal O_q$-analogue of Lusztig's divided powers; thesis Chapter 3 and Baseilhac-Gainutdinov-Vu 2017, Section 3.7).

(i) Presentation (the 'small quantum group' analog). Prove, or disprove, that the specialization of $\mathcal O_q$ at $q=e^{i\pi/N}$, generated by $W_0, W_1, W_0^{[N]}, W_1^{[N]}$, is presented by the four families of relations conjectured by Baseilhac-Gainutdinov-Vu (2017, relations (2.10), (4.20), (4.15) with its unnumbered companion, and (4.17)/(4.19); derived there for the class of cyclic tridiagonal pairs with the spectra of their Section 3, giving an algebra depending on the five parameters $N; b, c, b^*, c^*$): the $q$-Serre-type $q$-Dolan-Grady relations among $W_0, W_1$ evaluated at $q=e^{i\pi/N}$; the degree-$N$ polynomial relations satisfied by $W_0$ and $W_1$; the mixed relations between each $W_i$ and the divided polynomials; and the higher-order Onsager algebra relations satisfied by $W_0^{[N]}, W_1^{[N]}$ — that is, establish that these relations are complete defining relations (every relation obeyed by the four generators at $q=e^{i\pi/N}$ is a consequence of them).

(ii) Structure (the 'full quantum group' analog). Determine the algebraic structure of this algebra: whether it is finite-dimensional, its center, and whether it admits a Hopf-type or coideal/comodule structure analogous to Lusztig's full quantum group — in particular, whether coproduct-like formulas exist for the divided polynomials, or prove that no such structure exists.

(iii) Representation theory. Classify the finite-dimensional irreducible representations of this algebra (thesis 4.3, Problems 1-2); and, as a related but separate problem (thesis 4.2, Problem 1), classify the irreducible cyclic tridiagonal pairs up to isomorphism — pairs of diagonalizable linear maps $A, A^*$ on a finite-dimensional vector space, with no common invariant subspace, such that each acts on a cyclic ordering (indexed by $\mathbb Z_N$, with wrap-around $V_{-1}=V_{N-1}$, $V_0=V_N$; Baseilhac-Gainutdinov-Vu 2017, Definition 1.1) of the eigenspaces of the other block-tridiagonally. Because both operators of a cyclic tridiagonal pair are required to be diagonalizable, the pair classification addresses the diagonalizable sector and does not by itself cover possible non-semisimple irreducible modules with Jordan-cell action; neither classification is to be treated as completing the other. Also construct explicit cyclic tridiagonal pairs beyond those obtained from the open XXZ chain (for instance from other solutions of the reflection equation, the boundary analogue of the Yang-Baxter equation, or from multivariable orthogonal-polynomial operators; thesis 4.2, Problem 2).

(iv) Physical application. Characterize the boundary-parameter regimes and Hilbert-space sectors of the open XXZ spin chain at $q=e^{i\pi/N}$ in which a non-abelian symmetry associated with a subalgebra of the algebra generated by $W_0^{[N]}, W_1^{[N]}$ occurs and commutes with the Hamiltonian — the analogs of the loop-algebra symmetry sectors of the periodic chain at roots of unity, in which $\mathfrak{sl}_2\otimes\mathbb C[t,t^{-1}]$ (the Lie algebra of $\mathfrak{sl}_2$-valued Laurent polynomials) acts and commutes with the Hamiltonian. It is proved only in the special boundary-parameter case $k_-=0$ (thesis Section 3.3.3) that the Hamiltonian cannot commute simultaneously with both $W_0^{[N]}$ and $W_1^{[N]}$; the general sector question is left open there.

These four components jointly constitute one objective: the root-of-unity Lusztig analogy for the q-Onsager algebra, with the presentation (i) as its foundation, (ii)-(iii) as its structural and representation-theoretic content, and (iv) as its physical payoff. A complete solution requires all four components (i)-(iv), partial results being assessed component by component, so that resolving (i) alone settles only the presentation sub-question and leaves (ii)-(iv) open; if (i) is disproved, (ii)-(iv) are not thereby resolved and are to be settled for the algebra actually generated by $W_0, W_1, W_0^{[N]}, W_1^{[N]}$ at $q=e^{i\pi/N}$. The related but distinct open question of an r-th reflection-equation algebra $O_q^{(r)}$ and associated integrable hierarchies for $r\ge 2$ at generic q (thesis Perspective 4.1) is not part of this problem.

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