arXiv subjects
Publications and source records attributed to Takeo Kojima.
We study infinitely many commuting operators $T_B(z)$, which we call infinite transfer matrix of boundary $U_{q,p}(A_{N-1}^{(1)})$ face model. We diagonalize infinite transfer matrix $T_B(z)$ by using free field realizations of the vertex operators of the elliptic quantum group $U_{q,p}(A_{N-1}^{(1)})$.
We study the ground state of the boundary Izergin-Korepin model. The boundary Izergin-Korepin model is defined by so-called $R$-matrix and $K$-matrix for $U_q(A_2^{(2)})$ which satisfy Yang-Baxter equation and boundary Yang-Baxter equation respectively. The ground state associated with identity $K$-matrix $K(z)=id$ was constructed in earlier study [Yang and Zhang, Nucl.Phys.B596,495-(2001)]. We construct the free field realization of the ground state associated with nontrivial diagonal $K$-matrix.
We diagonalize infinitely many commuting operators $T_B(z)$. We call these operators $T_B(z)$ the boundary transfer matrix associated with the quantum group and the elliptic quantum group. The boundary transfer matrix is related to the solvable model with a boundary. When we diagonalize the boundary transfer matrix, we can calculate the correlation functions for the solvable model with a boundary. We review the free field approach to diagonalization of the boundary transfer matrix $T_B(z)$ associated with $U_q(A_2^{(2)})$ and $U_{q,p}(\hat{sl_N})$. We construct the free field realizations of the eigenvectors of the boundary transfer matrix $T_B(z)$. This paper includes new unpublished formula of the eigenvector for $U_q(A_2^{(2)})$. It is thought that this diagonalization method can be extended to more general quantum group $U_q(g)$ and elliptic quantum group $U_{q,p}(g)$.
We construct two classes of infinitely many commuting operators associated with the elliptic quantum group $U_{q,p}(\hat{sl_N})$. We call one of them the integral of motion ${\cal G}_m$, $(m \in {\mathbb N})$ and the other the boundary transfer matrix $T_B(z)$, $(z \in {\mathbb C})$. The integral of motion ${\cal G}_m$ is related to elliptic deformation of the $N$-th KdV theory. The boundary transfer matrix $T_B(z)$ is related to the boundary $U_{q,p}(\hat{sl_N})$ face model. We diagonalize the boundary transfer matrix $T_B(z)$ by using the free field realization of the elliptic quantum group, however diagonalization of the integral of motion ${\cal G}_m$ is open problem even for the simplest case $U_{q,p}(\hat{sl_2})$.
In this review, we study free field realizations of the Feigin-Odesskii algebra. We construct free field realizations of a pair of infinitely many commutative operators, associated with the elliptic quantum group $U_{q,p}(\widehat{sl_N})$.
We construct a free field realization of the ground state of the boundary transfer matrix for the $U_{q,p}(\hat{sl}_3)$ ABF model. Using this ground state and type-II vertex operator, we have a diagonalization of the boundary transfer matrix.
We construct a free field realization of the elliptic quantum algebra $U_{q,p}(\hat{sl_N})$ for arbitrary level $k \neq 0,-N$. We study Drinfeld current and the screening current associated with $U_{q,p}(\hat{sl_N})$ for arbitrary level $k$. In the limit $p \to 0$ this realization becomes $q$-Wakimoto realization for $U_q(\hat{sl_N})$.
We study a free field realization of the elliptic quantum algebra $U_{q,p}(sl_3^)$ for arbitrary level $k$. We give the free field realization of elliptic analogue of Drinfeld current associated with $U_{q,p}(sl_3^)$ for arbitrary level $k$. In the limit $p \to 0, q \to 1$ our realization reproduce Wakimoto realization for the affine Lie algebra $sl_3^$.
The q-oscillator representation for the Borel subalgebra of the affine symmetry $U_q(sl_N^)$ is presented. By means of this q-oscillator representation, we give the free field realizations of the Baxter's Q-operator $Q_j(t)$, $\bar{Q}_j(t)$ for the W-algebra $W_N$. We give the functional relations of the $T$-$Q$ operators, including the higher-rank generalization of the Baxter's $T$-$Q$ relation.
Based on the vertex-face correspondence, we give an algebraic analysis formulation of correlation functions of the $k\times k$ fusion eight-vertex model in terms of the corresponding fusion SOS model. Here $k\in Z_{>0}$. A general formula for correlation functions is derived as a trace over the space of states of lattice operators such as the corner transfer matrices, the half transfer matrices (vertex operators) and the tail operator. We give a realization of these lattice operators as well as the space of states as objects in the level $k$ representation theory of the elliptic algebra $U_{q,p}(\hat{sl}_2)$.
We construct a realization of the L-operator satisfying the RLL-relation of the face type elliptic quantum group B_{q,lambda}(A_2^(2)). The construction is based on the elliptic analogue of the Drinfeld currents of U_q(A_2^(2)), which forms the elliptic algebra U_{q,p}(A_2^(2)). We give a realization of the elliptic currents E(z), F(z) and K(z) as a tensor product of the Drinfeld currents of U_q(A_2^(2)) and a Heisenberg algebra. In the level-one representation, we also give a free field realization of the elliptic currents. Applying these results, we derive a free field realization of the U_{q,p}(A_2^(2))-analogue of the B_{q,lambda}(A_2^(2)) -intertwining operators. The resultant operators coincide with those of the vertex operators in the dilute A_L model, which is known to be a RSOS restriction of the A_2^(2) face model.
After reviewing the recent results on the Drinfeld realization of the face type elliptic quantum group B_{q,lambda}(sl_N^) by the elliptic algebra U_{q,p}(sl_N^), we investigate a fusion of the vertex operators of U_{q,p}(sl_N^). The basic generating functions Λ_j(z) (j=1,2,.. N-1) of the deformed W_N algebra are derived explicitly.
By using the elliptic analogue of the Drinfeld currents in the elliptic algebra U_{q,p}(\hat{sl}_N), we construct a L-operator, which satisfies the RLL-relations characterizing the face type elliptic quantum group B_{q,λ}(\hat{sl}_N). For this purpose, we introduce a set of new currents K_j(v) (1\leq j\leq N) in U_{q,p}(\hat{sl}_N). As in the N=2 case, we find a structure of U_{q,p}(\hat{sl}_N) as a certain tensor product of B_{q,λ}(\hat{sl}_N) and a Heisenberg algebra. In the level-one representation, we give a free field realization of the currents in U_{q,p}(\hat{sl}_N). Using the coalgebra structure of B_{q,λ}(\hat{sl}_N) and the above tensor structure, we derive a free field realization of the U_{q,p}(\hat{sl}_N)-analogue of B_{q,λ}(\hat{sl}_N)-intertwining operators. The resultant operators coincide with those of the vertex operators in the A_{N-1}^{(1)}-type face model.
We study the affine $A_{n-1}^{(1)}$ Toda fields with boundary reflection. Our approach is based on the free field approach. We construct free field realizations of the boundary state and its dual. For an application of these realizations, we present integral representations for the form factors of the local operators. In a limiting case $ρ\to \infty$, our integral representations reproduce those of form factors for the SU(n) invariant massive Thirring model with boundary reflection [nlin.SI/0010020,Int.J.Mod.Phys.A16,no.15 (2001)2665-2689].
We present an integral representation to the quantum Knizhnik-Zamolodchikov equation associated with twisted affine symmetry $U_q(A_2^{(2)})$ for massless regime $|q|=1$. Upon specialization, it leads to a conjectural formula for the correlation function of the Izergin-Korepin model in massless regime $|q|=1$. In a limiting case $q \to -1$, our conjectural formula reproduce the correlation function for the Izergin-Korepin model at xritical point $q=-1$.
We study the time and temperature dependent correlation functions for an impenetrable Bose gas with Neumann or Dirichlet boundary conditions $\langle ψ(x_1,0)ψ^\dagger(x_2,t)\rangle _{\pm,T}$. We derive the Fredholm determinant formulae for the correlation functions, by means of the Bethe Ansatz. For the special case $x_1=0$, we express correlation functions with Neumann boundary conditions $\langleψ(0,0)ψ^\dagger(x_2,t)\rangle _{+,T}$, in terms of solutions of nonlinear partial differential equations which were introduced in \cite{kojima:Sl} as a generalization of the nonlinear Schrödinger equations. We generalize the Fredholm minor determinant formulae of ground state correlation functions $\langleψ(x_1)ψ^\dagger(x_2)\rangle _{\pm,0}$ in \cite{kojima:K}, to the Fredholm determinant formulae for the time and temperature dependent correlation functions $\langleψ(x_1,0)ψ^\dagger(x_2,t)\rangle _{\pm,T}$, $t \in {\bf R}$, $T \geq 0$.
We consider the recently obtained integral representation of quantum Knizhnik-Zamolodchikov equation of level 0. We obtain the condition for the integral kernel such that these solutions satisfy three axioms for form factor á la Smirnov. We discuss the relation between this integral representation and the form factor of XXZ spin chain.