Abstract:
We review the main result of cond-mat/0503564. The Hamiltonian of the XXZ spin chain and the transfer matrix of the six-vertex model has the $sl_2$ loop algebra symmetry if the $q$ parameter is given by a root of unity, $q_0^{2N}=1$, for an integer $N$. We discuss the dimensions of the degenerate eigenspace generated by a regular Bethe state in some sectors, rigorously as follows: We show that every regular Bethe ansatz eigenvector in the sectors is a highest weight vector and derive the highest weight ${ar d}_k^{pm}$, which leads to evaluation parameters $a_j$. If the evaluation parameters are distinct, we obtain the dimensions of the highest weight representation generated by the regular Bethe state.

Abstract:
We discuss multivariable invariants of colored links associated with the $N$-dimensional root of unity representation of the quantum group. The invariants for $N>2$ are generalizations of the multi-variable Alexander polynomial. The invariants vanish for disconnected links. We review the definition of the invariants through (1,1)-tangles. When $(N,3)=1$ and $N$ is odd, the invariant does not vanish for the parallel link (cable) of the knot $3_1$, while the Alexander polynomial vanishes for the cable link.

Abstract:
Some features of integrable lattice models are reviewed for the case of the six-vertex model. By the Bethe ansatz method we derive the free energy of the six-vertex model. Then, from the expression of the free energy we show analytically the critical singularity near the phase transition in the anti-ferroelectric regime, where the essential singularity similar to the Kosterlitz-Thouless transition appears. We discuss the connection of the six-vertex model to the conformal field theory with c=1. We also introduce various exactly solvable models defined on two-dimensional lattices such as the chiral Potts model and the IRF models. We show that the six-vertex model has rich mathematical structures such as the quantum groups and the braid group. The graphical approach is emphasized in this review. We explain the meaning of the Yang-Baxter equation by its diagram. Furthermore, we can understand the defining relation of the algebraic Bethe ansatz by the graphical representation. We can thus easily translate formulas of the algebraic Bethe ansatz into those of the statistical models. As an illustration, we show explicitly how we can derive Baxter's expressions from those of the algebraic Bethe ansatz.

Abstract:
We discuss a conjecture that the twisted transfer matrix of the six-vertex model at roots of unity with some discrete twist angles should have the sl(2) loop algebra symmetry. As an evidence of this conjecture, we show the following mathematical result on a subalgebra of the sl(2) loop algebra, which we call a Borel subalgebra: any given finite-dimensional highest weight representation of the Borel subalgebra is extended into that of the sl(2) loop algebra, if the parameters associated with it are nonzero. Thus, if operators commuting or anti-commuting with the twisted transfer matrix of the six-vertex model at roots of unity generate the Borel subalgebra, then they also generate the sl(2) loop algebra. The result should be useful for studying the connection of the sl(2) loop algebra symmetry to the Onsager algebra symmetry of the superintegrable chiral Potts model.

Abstract:
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 $sl_2$ loop algebra. Here $q$ is related to the XXZ anisotropic coupling $\Delta$ by $\Delta=(q+q^{-1})/2$, and it is given by a root of unity, $q^{2N}=1$, for a positive integer $N$. We show that regular XXZ Bethe states are annihilated by the generators ${\bar x}_k^{+}$'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, ${\bar h}_k$'s. Here the loop algebra $U(L(sl_2))$ is generated by ${\bar x}_k^{\pm}$ and ${\bar h}_k$ for $k \in {\bf Z}$, which are the classical analogues of the Drinfeld generators of the quantum loop algebra $U_q(L(sl_2))$. A representation of $U(L(sl_2))$ is called highest weight if it is generated by a vector $\Omega$ which is annihilated by the generators ${\bar x}_k^{+}$'s and such that $\Omega$ is an eigenvector of the ${\bar h}_k$'s. We also discuss the classical analogue of the Drinfeld polynomial which characterizes the irreducible finite-dimensional highest weight representation of $U(L(sl_2))$.

Abstract:
We review the main result of cond-mat/0503564. The Hamiltonian of the XXZ spin chain and the transfer matrix of the six-vertex model has the $sl_2$ loop algebra symmetry if the $q$ parameter is given by a root of unity, $q_0^{2N}=1$, for an integer $N$. We discuss the dimensions of the degenerate eigenspace generated by a regular Bethe state in some sectors, rigorously as follows: We show that every regular Bethe ansatz eigenvector in the sectors is a highest weight vector and derive the highest weight ${\bar d}_k^{\pm}$, which leads to evaluation parameters $a_j$. If the evaluation parameters are distinct, we obtain the dimensions of the highest weight representation generated by the regular Bethe state.

Abstract:
For the one-dimensional XXX model under the periodic boundary conditions, we discuss two types of eigenvectors, regular eigenvectors which have finite-valued rapidities satisfying the Bethe ansatz equations, and non-regular eigenvectors which are descendants of some regular eigenvectors under the action of the SU(2) spin-lowering operator. It was pointed out by many authors that the non-regular eigenvectors should correspond to the Bethe ansatz wavefunctions which have multiple infinite rapidities. However, it has not been explicitly shown whether such a delicate limiting procedure should be possible. In this paper, we discuss it explicitly in the level of wavefunctions: we prove that any non-regular eigenvector of the XXX model is derived from the Bethe ansatz wavefunctions through some limit of infinite rapidities. We formulate the regularization also in terms of the algebraic Bethe ansatz method. As an application of infinite rapidity, we discuss the period of the spectral flow under the twisted periodic boundary conditions.

Abstract:
We review an algebraic method for constructing degenerate eigenvectors of the transfer matrix of the eight-vertex Cyclic Solid-on-Solid lattice model (8V CSOS model), where the degeneracy increases exponentially with respect to the system size. We consider the elliptic quantum group $E_{\tau, \eta}(sl_2)$ at the discrete coupling constants: $2N \eta = m_1 + i m_2 \tau$, where $N, m_1$ and $m_2$ are integers. Then we show that degenerate eigenvectors of the transfer matrix of the six-vertex model at roots of unity in the sector $S^Z \equiv 0$ (mod $N$) are derived from those of the 8V CSOS model, through the trigonometric limit. They are associated with the complete $N$ strings. From the result we see that the dimension of a given degenerate eigenspace in the sector $S^Z \equiv 0$ (mod $N$) of the six-vertex model at $N$th roots of unity is given by $2^{2S_{max}^Z/N}$, where $S_{max}^Z$ is the maximal value of the total spin operator $S^Z$ in the degenerate eigenspace.

Abstract:
For the integrable spin-s XXZ chain we express explicitly any given spin-$s$ form factor in terms of a sum over the scalar products of the spin-1/2 operators. Here they are given by the operator-valued matrix elements of the monodromy matrix of the spin-1/2 XXZ spin chain. In the paper we call an arbitrary matrix element of a local operator between two Bethe eigenstates a form factor of the operator. We derive all important formulas of the fusion method in detail. We thus revise the derivation of the higher-spin XXZ form factors given in a previous paper. The revised method has several interesting applications in mathematical physics. For instance, we express the spin-$s$ XXZ correlation function of an arbitrary entry at zero temperature in terms of a sum of multiple integrals.

Abstract:
We discuss an algebraic method for constructing eigenvectors of the transfer matrix of the eight vertex model at the discrete coupling parameters. We consider the algebraic Bethe ansatz of the elliptic quantum group $E_{\tau, \eta}(sl_2)$ for the case where the parameter $\eta$ satisfies $2 N \eta = m_1 + m_2 \tau $ for arbitrary integers $N$, $m_1$ and $m_2$. When $m_1$ or $m_2$ is odd, the eigenvectors thus obtained have not been discussed previously. Furthermore, we construct a family of degenerate eigenvectors of the XYZ spin chain, some of which are shown to be related to the $sl_2$ loop algebra symmetry of the XXZ spin chain. We show that the dimension of some degenerate eigenspace of the XYZ spin chain on $L$ sites is given by $N 2^{L/N}$, if $L/N$ is an even integer. The construction of eigenvectors of the transfer matrices of some related IRF models is also discussed.