The authors discuss the XXZ spin chain with certain boundary terms and the representation of quantum group SU q (2). It is shown that Bethe ansatz states are highest-weight states of SU q (2). With a generic q they construct the irreducible representations of the quantum group. For q being a root of unity, they show that there are new Bethe ansatz states, which coincide with null states of SU q (2). By taking certain limits the authors can derive the state mod b), which is necessary for constructing the indecomposable but reducible representations of SU q (2) and for the completeness of the state space. In this case the Hamiltonian may not be completely diagonalized.
The formulation and resolution of integrable lattice statistical models in a quantum group covariant way is the subject of this review. The Bethe Ansatz turns to be remarkably useful to implement quantum group symmetries and to provide quantum group representations even when
q is a root of unity. We start by solving the six-vertex model with fixed boundary conditions (FBC) that guarantee exact
SU(2)q invariance on the lattice. The algebra of the Yang-Baxter (YB) and
SU(2)q generators turns to close. The infinite spectral parameter limit of the YB generators yields {\bf cleanly} the
SU(2)q generators. The Bethe Ansatz states constructed for FBC are shown to be {\bf highest weights} of
SU(2)q. The higher level Bethe Ansatz equations (BAE, describing the physical excitations) are explicitly derived for FBC. We then solve the RSOS(
p) models on the light--cone lattice with fixed boundary conditions by disentangling the type II representations of
SU(2)q, at
q=eiπ/p, from the full SOS spectrum obtained through Algebraic Bethe Ansatz. The RSOS states are those with quantum spin $J
We prove some part of the conjecture that regular Bethe ansatz eigenvectors of the XXZ spin chain at roots of unity are highest weight vectors of the
sl2 loop algebra. Here
q is related to the XXZ anisotropic coupling
Δ by
Δ=(q+q−1)/2, and it is given by a root of unity,
q2N=1, for a positive integer
N. We show that regular XXZ Bethe states are annihilated by the generators
xˉk+'s, for any
N. We discuss, for some particular cases of N=2, that regular XXZ Bethe states are eigenvectors of the generators of the Cartan subalgebra,
hˉk's. Here the loop algebra
U(L(sl2)) is generated by
xˉk± and
hˉk for
k∈Z, which are the classical analogues of the Drinfeld generators of the quantum loop algebra
Uq(L(sl2)). A representation of
U(L(sl2)) is called highest weight if it is generated by a vector
Ω which is annihilated by the generators
xˉk+'s and such that
Ω is an eigenvector of the
hˉk's. We also discuss the classical analogue of the Drinfeld polynomial which characterizes the irreducible finite-dimensional highest weight representation of
U(L(sl2)).
Baxter's Q-operator for the quantum transfer matrix of the XXZ spin-chain is constructed employing the representation theory of quantum groups. The spectrum of this Q-operator is discussed and novel functional relations which describe the finite temperature regime of the XXZ spin-chain are derived. For non-vanishing magnetic field the previously known Bethe ansatz equations can be replaced by a system of quadratic equations which is an important advantage for numerical studies. For vanishing magnetic field and rational coupling values it is argued that the quantum transfer matrix exhibits a loop algebra symmetry closely related to the one of the classical six-vertex transfer matrix at roots of unity.