Universal TT- and TQ-relations via centrally extended q-Onsager algebra
Pascal Baseilhac, Azat M. Gainutdinov, Guillaume Lemarthe
TL;DR
This work develops a representation-theoretic, universal framework for open quantum spin chains based on the centrally extended q-Onsager algebra Aq. By constructing universal spin-j transfer matrices ${\bf {\mathsf{T}}}^{(j)}(u)$ from fused K-operators and dual K-matrices within Sklyanin's reflection algebra, it establishes a general TT-relations hierarchy (Theorem TTrel) that governs all spins and boundary conditions. The TT-relations yield explicit T-systems, Y-systems, and universal TQ-relations for Aq and its degenerate quotients, and they underlie a concrete algorithm to compute all local conserved quantities as polynomials in a finite set of nonlocal Aq generators, with clear spin-chain realizations via the two generators ${\mathcal W}_0^{(N)},{\mathcal W}_1^{(N)}$. The framework also reveals exchange relations and emergent symmetries of spin-j Hamiltonians under specific boundary parameter regimes, and it extends naturally to truncated algebras ${\mathcal A}_q^{(N)}$ and to diagonal boundary cases, connecting with known results (XXZ/XXX) and forming a bridge to T- and Y-systems and TQ-relations in broader integrable contexts.
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.