We develop the Baxterization approach to (an extension of) the quantum group GL_q(2). We introduce two matrices which play the role of spectral parameter dependent L-matrices and observe that they are naturally related to two different comultiplications. Using these comultiplication structures, we find the related fundamental R-operators in terms of powers of coproducts and also give their equivalent forms in terms of quantum dilogarithms. The corresponding quantum local Hamiltonians are given in terms of logarithms of positive operators. An analogous construction is developed for the q-oscillator and Weyl algebras using that their algebraic and coalgebraic structures can be obtained as reductions of those for the quantum group. As an application, the lattice Liouville model, the q-DST model, the Volterra model, a lattice regularization of the free field, and the relativistic Toda model are considered.
Deep Dive into Baxterization of GL_q(2) and its application to the Liouville model and some other models on a lattice.
We develop the Baxterization approach to (an extension of) the quantum group GL_q(2). We introduce two matrices which play the role of spectral parameter dependent L-matrices and observe that they are naturally related to two different comultiplications. Using these comultiplication structures, we find the related fundamental R-operators in terms of powers of coproducts and also give their equivalent forms in terms of quantum dilogarithms. The corresponding quantum local Hamiltonians are given in terms of logarithms of positive operators. An analogous construction is developed for the q-oscillator and Weyl algebras using that their algebraic and coalgebraic structures can be obtained as reductions of those for the quantum group. As an application, the lattice Liouville model, the q-DST model, the Volterra model, a lattice regularization of the free field, and the relativistic Toda model are considered.
A quantum model is a system (H, A, H), with a Hilbert space H, an algebra of observables A, and a Hamiltonian H. The model is integrable if there exists a complete set of quantum integrals of motion, i.e., a set of self-adjoint elements of A which commute with each other and with the Hamiltonian. For homogeneous one-dimensional lattice models one has H = K ⊗N , A = B ⊗N , with one copy of Hilbert space K and algebra of local observables B being associated to each of the N sites of a one-dimensional lattice. K is usually characterized as a representation of an algebra U of "symmetries", and B is generated from the operators which represent the elements of U on K.
A key step in constructing an integrable lattice model is to find an L-matrix L(λ) ∈ Mat(n) ⊗ B and an auxiliary R-matrix R(λ) ∈ Mat(n) ⊗2 such that the following matrix commutation relation
where λ, µ ∈ C, is equivalent to the defining relations of U. Here and below we use the standard notation: subscripts indicate nontrivial components in tensor product, e.g., R 12 ≡ R ⊗ 1, etc.
For further details on the R-matrix approach to quantum integrability we refer the reader to the review [F1].
For a model on a closed one-dimensional lattice, i.e., with periodic boundary conditions, a set of quantum integrals of motion is generated by the auxiliary transfer-matrix T (λ) = tr a L a,N (λ) . . . L a,1 (λ) . However, these integrals are in general non-local, i.e., they are not representable as a sum of terms each containing nontrivial contributions only from several nearest sites. The recipe [FT2] for constructing local integrals of motion for a model with a given L-matrix is to find first the corresponding fundamental R-operator R(λ) ∈ B ⊗2 , which satisfies the following intertwining relation (here and below we will use it in the braid form): R 23 (λ) L 12 (λµ) L 13 (µ) = L 12 (µ) L 13 (λµ) R 23 (λ) .
(2)
The corresponding transfer-matrix is constructed as T(λ) = tr a R aN (λ)P aN . . . R a1 (λ)P a1 , where the subscript a stands now for an auxiliary copy of B, and P is the unitary operator permuting tensor factors in B ⊗2 . The fundamental R-operator is usually regular, that is, after appropriate normalization, it satisfies the relation
If the regularity condition holds, then first and higher order logarithmic derivatives of T(λ) at λ=1 are local integrals of motion for the periodic homogeneous model in question. In particular, the Hamiltonian is often chosen as the most local integral which involves only nearest neighbour interaction:
where the summation assumes that N+1 ≡ 1.
Thus, finding the fundamental R-operator for a given L-matrix is an important part of the Rmatrix approach to quantum integrable models. Furthermore, this problem is closely related to the problem of constructing the corresponding evolution operators and Q-operators. However, there is no general method for solving equation (2). The particular difficulty here is that it is not clear apriori on which operator argument(s) the function R(λ) depends.
Among the few known examples of constructing a fundamental R-operator the most algebraically transparent are those related to the case where the symmetry U admits the structure of a bialgebra. Such examples include the XXX spin chain [KRS] and closely related nonlinear Schrödinger model [FT2], where U = U (sl 2 ); and the XXZ spin chain [J1] and closely related sine-Gordon model [FT2,T1], where U = U q (sl 2 ). A crucial observation for solving (2) in these cases is that the operator argument of R(λ) is ∆(C q ), where C q is the Casimir element of U and ∆ is the comultiplication that defines the bialgebra structure of U. The corresponding solutions to (2) are expressed, respectively, in terms of the Gamma function or its q-analogue (see [J1, T1, F1, B2] for more details in the latter case).
The aim of the present article is to develop a similar algebraic construction of fundamental R-operators for models whose underlying symmetry corresponds, in the sense of Eq. ( 1), to the quantum group GL q (2). More precisely, we introduce the quantum group GL q (2) with generators a, b, c, d, θ, where θ may be chosen to be the inverse to b or c. It will be important to consider special positive representations of GL q (2) which ensures that the operators that we use are positive self-adjoint. These properties are crucial for constructing fundamental R-operators since we will need non-polynomial functions of generators and their coproducts.
The article is organized as follows. First, we discuss Baxterization of GL q (2) and GL q (2), presenting their defining relations in the form (1). The two matrices, g(λ) and ĝ(λ), which play the role of an L-matrix for GL q (2), will be our main objects of consideration. Next, we show that, besides the standard comultiplication ∆, there is another algebra homomorphism δ : GL q (2) → GL q (2) ⊗2 . Further, we solve Eq. (2) for g(λ) and ĝ(λ). The corresponding fundamental R-operators are given (up to some tw
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