Universal TT- and TQ-relations via centrally extended q-Onsager algebra

Abstract: Let $A_q$ be the alternating central extension of the q-Onsager algebra, a comodule algebra over the quantum loop algebra of $sl_2$. We first classify one-dimensional representations of $A_q$, on which spin-j K-operators constructed in [LBG] act as K-matrices. Using the K-operators and these K-matrices, we construct universal spin-j transfer matrices generating commutative subalgebras in $A_q$. Within a technical conjecture, we derive their fusion hierarchy, the so-called universal TT-relations. On spin-chain representations of $A_q$, we show how the universal transfer matrices evaluate to spin-chain transfer matrices, and as a result we get explicit TT-relations for all values of spins for auxiliary and quantum spaces, any inhomogeneities, and general integrable boundary conditions. In particular, we derive previously conjectured TT-relations. Using the TT-relations, we show that n-th local conserved quantities of the spin-j chains of length N are polynomials of total degree 4Njn in two non-local operators of the q-Onsager algebra. As a result, we give an algorithm of explicit calculation of all conserved quantities (Hamiltonians and higher logarithmic derivatives of the transfer matrix) in terms of spin operators. Furthermore, using the universal TT-relations we derive exchange relations between spin-j Hamiltonians and the two non-local operators, which shows existence of non-trivial symmetries for special boundary conditions, in the sense that they commute with all Hamiltonian densities. As a yet another application of our universal TT-relations we propose universal T-system, Y-system and universal TQ-relations for $A_q$, and as a result, universal TQ for the q-Onsager algebra. In view of application to diagonal boundary conditions, we also obtain universal TT- and TQ-relations for a certain degenerate version of $A_q$ known as centrally extended augmented q-Onsager algebra.

• The paper presents a novel derivation of universal TT-relations for open spin chains within the centrally extended q-Onsager algebra framework.
• It utilizes fusion hierarchies and quantum determinants to construct generating functions for transfer matrices, crucial for solving complex Hamiltonians.
• The approach extends Bethe Ansatz methods and links to blob algebra, highlighting practical implications for quantum integrable models with boundaries.

Universal TT- and TQ-relations via Centrally Extended q-Onsager Algebra

Introduction

This paper addresses integral aspects of the algebraic structures in quantum integrable spin chains, particularly through the framework of the $q$-Onsager algebra and its central extensions. The emphasis is on developing and exploring universal transfer matrix $^{(j)}(u)$ formulations and associated TT-relations in the context of the centrally extended $q$-Onsager algebra, $\mathcal{A}_q$.

These constructs form essential components in the broader scope of solving quantum integrable models, particularly those with boundaries, by explicitly constructing generating functions of commuting elements derived from the fusion hierarchy and Baxter's TQ-relations. This is pertinent to both equilibrium and non-equilibrium physics, offering potential pathways to solving models' Hamiltonians down to their spectral level.

Universal TT-relations and Their Derivation

The paper introduces the universal TT-relations as a way to encapsulate the interrelations in the big family of transfer matrices characterizing open spin chains. The representation theory underpinning closes on the $q$-Onsager algebra, with universal TT-relations expressed as: $$^{(j)}(u) = ^{(j-1)}(u q^{-1}) ^{(1)}(u q^{j-1}) + \frac{\Gamma (u q^{j-3}) \Gamma_+ (uq^{j-3})}{ c(u^2 q^{2j}) c(u^2 q^{2j-2}) } ^{(j-1)}(u q^{-1}).$$ These TT-relations hold for integer and half-integer $j$, assuming a technical conjecture about K-operators.

The proof leverages a detailed understanding of $\mathcal{A}_q$, utilizing key aspects such as:

• Coprincipal Elements: Roles are played by quantum determinants $\Gamma(u)$ and $\Gamma_+(u)$, instrumental in capturing spectral properties.
• Fusion Hierarchies: Progressive construction of higher-spin transfer matrices through recursion relations and a fusion technique.

Implementations and Implications

The methodology scopes beyond mere theoretical formulations; it encompasses:

1. Spin-chain Representations and Hamiltonians: Through limiting operations and parameter substitutions, the theoretical models align with empirical models like the XXZ spin chain. Hamiltonian formulations obtained through logarithmic derivatives of transfer matrices offer real applications in analyzing quantum spin dynamics.
2. Bethe Ansatz and Tridiagonal Pair Extensions: Integrability is sought through Bethe-type solutions, yet the paradigm here gets an extension in tridiagonal pair theories, facilitating the treatment of complex boundary conditions in spin models.
3. Relations to Blob Algebra and Mathematical Structures: These mathematical constructs entwine with the blob algebra, emphasizing the algebraic symmetries governing spin-chain dynamics.

Conclusion

The exploration of the universal TT- and TQ-relations within the centrally extended $q$-Onsager algebra $\mathcal{A}_q$ opens vistas in the analysis of quantum integrable models. The derivation and applications thereof—whether in constructing exact solutions or understanding deep algebraic symmetries—underscore the complexity and practicality of such mathematical frameworks. Moreover, consequent T-system and Y-system for $\mathcal{A}_q$ emphasize the algebra's cohesiveness in encoding integrable structures, beneficial in theoretical physics and applicable computational scenarios.