Generalized Onsager algebras: irreducible modules beyond one-dimensional characters and the classified finite-dimensional cases, and inhomogeneous Serre presentations beyond symmetrizable Cartan matrices

The Onsager algebra, the Lie algebra underlying the commuting transfer matrices of Onsager's 1944 solution of the two-dimensional Ising model, is isomorphic to the fixed-point subalgebra of the affine Lie algebra of type $A_1^{(1)}$ under its Chevalley involution. Stokman (2020) generalized this: for every symmetrizable generalized Cartan matrix $A$, the fixed-point Lie subalgebra $k(A)$ of the Kac-Moody algebra $g(A)$ under the Chevalley involution admits a presentation by generators $B_1,\dots,B_n$ subject to inhomogeneous Serre relations with explicitly recursively defined coefficients. What remains open are two directions extending the paper's results and hypotheses beyond the settled regime: (1) the representation theory of the generalized Onsager algebra $k(A)$ beyond its one-dimensional characters and beyond the finite-dimensional modules already classified in finite type (where $k(A)$ is reductive) and untwisted affine type (where $k(A)$ is a twisted loop algebra of the second kind, an equivariant map algebra), in particular the infinite-dimensional irreducible modules, for which no classification exists even for the original Onsager algebra, and anything at all for indefinite Cartan type; and (2) inhomogeneous Serre-type presentations beyond the symmetrizable Chevalley setting: non-symmetrizable generalized Cartan matrices, where not even the ambient Kac-Moody algebra admits a known finite Serre presentation, and explicit classical presentations for general involutions of the second kind, where the quantum side is settled but the classical limit has been written out only in the split case.

Background

The Onsager algebra $O$ is the complex Lie algebra with basis $A_k$ ($k \in \mathbb{Z}$) and $G_m$ ($m \geq 1$) and brackets $[A_k, A_l] = G_{l-k}$, $[G_m, G_n] = 0$, $[G_m, A_k] = 2(A_{k+m} - A_{k-m})$, with the convention $G_{-m} = -G_m$ for $m \geq 1$ and $G_0 = 0$. It appeared in Onsager's 1944 solution of the two-dimensional Ising model in zero field, where it accounts for the commutativity of the transfer matrices, the matrices whose ordered product along a lattice direction builds the partition function of the model. Dolan and Grady (1982) showed that $O$ is generated by two elements $B_0, B_1$ satisfying the cubic relations $[B_0,[B_0,[B_0,B_1]]] = -4[B_0,B_1]$ and $[B_1,[B_1,[B_1,B_0]]] = -4[B_1,B_0]$; these Dolan-Grady relations are the prototype of the presentations studied here. A classical result due to Roan and recorded in the source paper identifies $O$ with the fixed-point subalgebra of the affine Lie algebra of type $A_1^{(1)}$ under its Chevalley involution.

A generalized Cartan matrix (GCM) is an integral $n \times n$ matrix $A = (a_{ij})$ with $a_{ii} = 2$, $a_{ij} \leq 0$ for $i \neq j$, and $a_{ij} = 0$ exactly when $a_{ji} = 0$; it is symmetrizable if $DA$ is symmetric for some diagonal matrix $D$ with positive diagonal entries. The Kac-Moody algebra $g(A)$ is the Lie algebra generated by a Cartan subalgebra $h(A)$ and Chevalley generators $e_i, f_i$ ($1 \leq i \leq n$) subject to the relations $[h, h'] = 0$, $[h, e_j] = \alpha_j(h) e_j$, $[h, f_j] = -\alpha_j(h) f_j$ and $[e_i, f_j] = \delta_{ij} h_i$ for $h, h' \in h(A)$, where the pairs $(\alpha_j, h_j) \in h(A)^* \times h(A)$ realize $A$ through $\alpha_j(h_i) = a_{ij}$; for arbitrary $A$ one divides by the maximal ideal having trivial intersection with $h(A)$. A theorem of Gabber and Kac says that for symmetrizable $A$ this quotient is already presented by the generating relations together with the Serre relations $(ad\ e_i)^{1-a_{ij}} e_j = 0 = (ad\ f_i)^{1-a_{ij}} f_j$ ($i \neq j$). For non-symmetrizable $A$ no finite presentation of $g(A)$ of this type is known. The Chevalley involution $\omega$ of $g(A)$ is the involutive automorphism with $\omega(e_i) = -f_i$, $\omega(f_i) = -e_i$, $\omega(h) = -h$ for $h \in h(A)$; it is the simplest involution of the second kind, i.e. an involutive automorphism acting as $-1$ on the Cartan subalgebra.

Stokman (2020) defined for symmetrizable $A$ the generalized Onsager algebra $L(A)$: the Lie algebra with generators $B_1, \dots, B_n$ and, for $i \neq j$, the inhomogeneous Serre relations

$\sum_{s=0}^{1-a_{ij}} c_s^{ij}[1-a_{ij}]\ (ad\ B_i)^s B_j = 0,$

where the integers $c_s^{ij}[r]$ are fixed by a linear recursion in $r$, with coefficients quadratic in $r$ and depending on $a_{ij}$; for $a_{ij} = -2$ this is exactly a Dolan-Grady relation. The main theorem (Theorem 2.7) states that $L(A) \cong k(A) := g(A)^\omega$ for all symmetrizable $A$. The paper also determined all one-dimensional representations of $k(A)$ for every symmetrizable $A$ (they form a space of dimension $|E_A|$, where $E_A$ indexes the columns of $A$ whose entries are all even), computed integral forms and Chevalley-type structure constants in finite and untwisted affine type, recovering Onsager's original presentation, and analysed the symplectic types in detail. In finite type $k(A)$ is reductive (for example type $C_r$ gives $gl_r$), so its finite-dimensional representation theory is classical. In untwisted affine type $k(A)$ is a twisted loop algebra of the second kind: the fixed points of $g \otimes \mathbb{C}[t, t^{-1}]$ under $X \otimes t^n \mapsto \omega_0(X) \otimes t^{-n}$, where $g$ is the underlying finite-dimensional simple Lie algebra and $\omega_0$ its Chevalley involution. Such algebras are equivariant map algebras, meaning Lie algebras of equivariant regular maps from a variety carrying a finite automorphism group into a finite-dimensional Lie algebra, and the finite-dimensional irreducible modules of arbitrary equivariant map algebras were classified by Neher, Savage and Senesi.

The quantum analogue replaces the pair $(g(A), k(A))$ by a quantum symmetric pair $(U_q(g(A)), B_{c,s})$: here $U_q(g(A))$ is the quantized enveloping algebra, the Hopf-algebra deformation of the universal enveloping algebra of $g(A)$, and $B_{c,s}$ is a one-sided coideal subalgebra of $U_q(g(A))$, meaning a subalgebra whose image under the coproduct $\Delta$ satisfies $\Delta(B_{c,s}) \subseteq B_{c,s} \otimes U_q(g(A))$. The algebras $B_{c,s}$ are parametrized by combinatorial data called admissible pairs, which encode involutions of the second kind up to conjugacy; these coideals are called ıquantum groups (the dotless ı marks the involution), and the split case containing the Chevalley involution yields the generalized q-Onsager algebras introduced by Baseilhac and Belliard in connection with integrable lattice models and boundary affine Toda field theories. At the time of the source paper, explicit defining relations for $B_{c,s}$ were known only when all off-diagonal Cartan integers satisfy $a_{ij} \geq -3$ (building on the finite-type theory; see the account in Kolb's 2014 paper), and this bottleneck motivated the paper's stated hope for a complete and explicit algebraic presentation of Kolb's coideal subalgebras. That hope has since been fulfilled: explicit defining relations for $B_{c,s}$ are now known for all admissible pairs (Chen, Lu and Wang 2021; De Clercq 2021; Kolb and Yakimov 2021).

Problem Statement

Let $A$ be a symmetrizable generalized Cartan matrix, $g(A)$ the associated Kac-Moody algebra, $\omega$ its Chevalley involution, and $k(A) = g(A)^\omega \cong L(A)$ the generalized Onsager algebra with its inhomogeneous Serre presentation (Stokman, 2020). The following two independent open problems extend the paper's results and hypotheses beyond the settled regime.

(1) Representation theory of $k(A)$ beyond one-dimensional characters and the already classified finite-dimensional modules. Develop the module theory of the Onsager algebra ($A$ of untwisted affine type $A_1^{(1)}$) and of its higher-rank analogues $k(A)$ outside the settled cases, in particular: (a) the infinite-dimensional irreducible modules, for which no classification or structural parametrization (by highest-weight-type, evaluation-type or any other normal form) is known in the literature; (b) all irreducible modules for symmetrizable $A$ of indefinite type (neither finite nor affine), where $k(A)$ is not an equivariant map algebra and nothing beyond the one-dimensional characters is known; and (c) the category-theoretic structure of these module categories, where any categorical structural result about them, for example a determination of their blocks, extensions, dualities or tensor structure, counts as a resolution of this sub-item.

(2) Inhomogeneous Serre presentations beyond the symmetrizable Chevalley setting. (a) For a non-symmetrizable generalized Cartan matrix $A$, determine whether the fixed-point subalgebra $k(A) = g(A)^\omega$, generated by the elements $B_i = e_i - f_i$, admits a presentation by explicit inhomogeneous Serre-type relations. In this regime even the ambient algebra $g(A)$ has no known finite presentation (the Gabber-Kac theorem requires symmetrizability), so the correct formulation of the relations is itself part of the problem. (b) As a secondary item, write out explicitly the classical ($q \to 1$) inhomogeneous Serre presentations of the fixed-point subalgebras $g(A)^\theta$ for general involutions $\theta$ of the second kind (general admissible pairs), where the quantum-side presentations are known but the classical limit has been carried out only in the split (Chevalley) case.

The common objective of the two parts is to complete the structural theory of generalized Onsager algebras, presentations and representation theory, outside the regime in which it is already settled. The two parts are logically independent, and both are required to complete the common objective. Each part is independently resolvable: a complete resolution of a part requires all of its sub-items except the optional one, while resolving any single sub-item is a contribution toward that part. Sub-item (2)(b) is optional; every other sub-item is mandatory.

References

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