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

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 classify one-dimensional representations of $A_q$, and show that spin-j K-operators constructed in arXiv:2301.00781 act as K-matrices previously obtained in the literature. Using these K-operators and 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 local conserved quantities 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 showing non-trivial symmetries for special boundary conditions, that they commute with all Hamiltonian densities. As 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. For diagonal boundary conditions, we also obtain universal TT- and TQ-relations for a degenerate version of $A_q$ known as centrally extended augmented q-Onsager algebra. We finally discuss implications of our results for generalized Gibbs ensemble construction.