Universal Baxter TQ-relations for open boundary quantum integrable systems

Tsuboi, Zengo

doi:10.48550/arxiv.2010.09675

Cited by 1 publication

(2 citation statements)

References 57 publications

(137 reference statements)

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“…The K-operators (and their cousins) also play a crucial role in the spectrum analysis of the corresponding quantum spin chains, for instance, in the diagonalization of the Hamiltonian of the half-infinite XXZ spin-chain with diagonal boundary conditions [JKKKM94] (see also [BB12] in Onsager's approach) and with triangular ones [BB12], or describe hidden non-abelian symmetries [BB16] of the model. The K-operators are also used to construct the Baxter's Q-operator for diagonal boundary conditions [BT17, VW20] and triangular boundary conditions [Ts19,Ts20]. For all these models, the transfer matrix is the image in the spin-chain representation of a generating function t ( 1 2 ) (u) built from the K-operator (5.4) and a dual solution of the reflection equation for a spin-1 2 auxiliary space.…”

Section: Discussionmentioning

confidence: 99%

“…• Let us mention that the construction of a universal Q-operator for A q and corresponding T Q-relations may be also addressed. In that case, it would be desirable to construct the analogue of the fused Koperator for j → ∞ as suggested in [YNZ05], see also [VW20,Ts20]. • Finally, it is very desirable to construct K-operators of arbitrary complex spins, i.e., associated to U q sl 2 Verma modules of complex weights, as it would give essentially the corresponding universal K-matrix.…”

Section: Discussionmentioning

confidence: 99%

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Fused K-operators and the $q$-Onsager algebra

Guillaume¹,

Baseilhac²,

Gainutdinov³

2023

Preprint

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We study universal solutions to reflection equations with a spectral parameter, so-called Koperators, within a general framework of universal K-matrices -an extended version of the approach introduced by Appel-Vlaar. Here, the input data is a quasi-triangular Hopf algebra H, its comodule algebra B and a pair of consistent twists. In our setting, the universal K-matrix is an element of B ⊗ H satisfying certain axioms, and we mostly consider the case H = LUqsl 2 , the quantum loop algebra for sl 2 , and B = Aq, the alternating central extension of the q-Onsager algebra. Considering tensor products of evaluation representations of LUqsl 2 in "non-semisimple" cases, the new set of axioms allows us to introduce and study fused K-operators of spin-j; in particular, to prove that for all j ∈ 1 2 N they satisfy the spectral-parameter dependent reflection equation. We provide their explicit expression in terms of elements of the algebra Aq for small values of spin-j. The precise relation between the fused K-operators of spin-j and evaluations of a universal K-matrix for Aq is conjectured based on supporting evidences. Independently, we study K-operator solutions of the twisted intertwining relations associated with the comodule algebra Aq, and expand them in the Poincaré-Birkhoff-Witt basis of Aq. With a reasonably general ansatz, we found a unique solution for first few values of j which agrees with the fused K-operators, as expected. We conjecture that in general such solutions are uniquely determined and match with the expressions of the fused K-operators.

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