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arXiv:1401.0474 · Journal of Physics A Mathematical and Theoretical

The universalR-matrix and factorization of the L-operators related to the Baxter Q-operators

S. Khoroshkin, Zengo Tsuboi

Published Apr 25, 2014• 17 citations
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Abstract

We consider the 'universal monodromy operators' for the Baxter Q-operators. They are given as images of the universal R -matrix in oscillator representation. We find related universal factorization formulas in the case.
DOI
10.1088/1751-8113/47/19/192003

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We develop an approach for constructing the Baxter -operators for generic sl ( N ) spin chains. The key element of our approach is the possibility of representing a solution of the Yang–Baxter equation in the factorized form. We prove that such a representation holds for a generic sl ( N ) invariant -operator and find the explicit expression for the factorizing operators. Taking trace of monodromy matrices constructed of the factorizing operators one defines a family of commuting (Baxter) operators on the quantum space of the model. We show that a generic transfer matrix factorizes into the product of N Baxter -operators and discuss an application of this representation for a derivation of functional relations for transfer matrices.
Abstract One of the features of Baxter’s Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang–Baxter equation associated to particular infinite-dimensional representations). To extend such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang–Baxter equation) associated to these representations. In the case of quantum affine sl2\mathfrak {sl}_2sl2​ sl 2 and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.
In the present paper we describe the procedure of the Q-operators construction for the q-deformed model, described by the Lax operator, which is important to formulate the Bethe ansatz for the Sin-Gordon model. This Lax operator can also be considered as some massless limit of the Lax operator of SG model. We constructed two R-operators which are the universal intertwiners for the Lax operators. The traces of its monodromies over the auxiliary space are Baxter operators i.e. the operator solutions of T-Q equation. We also found the intertwining relations which imply the mutual commutativity of the corresponding Q-operators.
By the universal integrability objects we mean certain monodromy-type and transfer- type operators, where the representation in the auxiliary space is properly fixed, while the representation in the quantum space is not. This notion is actually determined by the structure of the universal R-matrix. We call functional relations between such universal integrability objects, and so, being independent of the representation in the quantum space, the universal functional relations. We present a short review of the universal functional relations for the quantum integrable systems associated with the quantum groups of loop Lie algebras.
In this paper, we construct a Q -operator as a trace of a representation of the universal R -matrix of U q over an infinite-dimensional auxiliary space. This auxiliary space is a four-parameter generalization of the q -oscillator representations used previously. We derive generalized T – Q relations in which three of these parameters shift. After a suitable restriction of parameters, we give an explicit expression for the Q -operator of the 6-vertex model and show the connection with Baxter's expression for the central block of his corresponding operator.