Quantum affine symmetry in vertex models

We systematise and develop a graphical approach to the investigations of quantum integrable vertex statistical models and the corresponding quantum spin chains. The graphical forms of the unitarity and various crossing relations are introduced. Their explicit analytical forms for the case of integrable systems associated with the quantum loop algebra ${\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))$ are given. The commutativity conditions for the transfer operators of lattic...

The infinite configuration space of an integrable vertex model based on $U_q\bigl(\hat{gl}(2|2)\bigr)_1$ is studied at $q=0$. Allowing four particular boundary conditions, the infinite configurations are mapped onto the semi-standard supertableaux of pairs of infinite border strips. By means of this map, a weight-preserving one-to-one correspondence between the infinite configurations and the normal forms of a pair of creation algebras is established for one boundary conditio...

The diagonalisation of the transfer matrices of solvable vertex models with alternating spins is given. The crystal structure of (semi-)infinite tensor products of finite-dimensional $U_q(\hat{sl}_2)$ crystals with alternating dimensions is determined. Upon this basis the vertex models are formulated and then solved by means of $U_q(\hat{sl}_2)$ intertwiners.

We introduce and solvev a special family of integrable interacting vertex models that generalizes the well known six-vertex model. In addition to the usual nearest-neighbor interactions among the vertices, there exist extra hard-core interactions among pair of vertices at larger distances.The associated row-to-row transfer matrices are diagonalized by using the recently introduced matrix product {\it ansatz}. Similarly as the relation of the six-vertex model with the XXZ quan...

$q$-vertex operators for quantum affine algebras have played important role in the theory of solvable lattice models and the quantum Knizhnik-Zamolodchikov equation. Explicit constructions of these vertex operators for most level one modules are known for classical types except for type $C_n^{(1)}$, where the level -1/2 have been constructed. In this paper we survey these results for the quantum affine algebras of types $A_n^{(1)}, B_n^{(1)}, C_n^{(1)}$ and $D_n^{(1)}$.

Drinfeld realisations are constructed for the quantum affine superalgebras of the series ${\rm\mathfrak{osp}}(1|2n)^{(1)}$,${\rm\mathfrak{sl}}(1|2n)^{(2)}$ and ${\rm\mathfrak{osp}}(2|2n)^{(2)}$. By using the realisations, we develop vertex operator representations and classify the finite dimensional irreducible representations for these quantum affine superalgebras.

We give explicit constructions of quantum symplectic affine algebras at level 1 using vertex operators.

We extend the recently developed Izergin-Korepin analysis on the wavefunctions of the $U_q(sl_2)$ six-vertex model to the reflecting boundary conditions. Based on the Izergin-Korepin analysis, we determine the exact forms of the symmetric functions which represent the wavefunctions and its dual. Comparison of the symmetric functions with the coordinate Bethe ansatz wavefunctions for the open XXZ chain by Alcaraz-Barber-Batchelor-Baxter-Quispel is also made. As an application,...

We construct the explicit $Q$-operator incorporated with the $sl_2$-loop-algebra symmetry of the six-vertex model at roots of unity. The functional relations involving the $Q$-operator, the six-vertex transfer matrix and fusion matrices are derived from the Bethe equation, parallel to the Onsager-algebra-symmetry discussion in the superintegrable $N$-state chiral Potts model. We show that the whole set of functional equations is valid for the $Q$-operator. Direct calculations...

The Algebraic Bethe ansatz for a supersymmetric nineteen vertex-model constructed from a three-dimensional representation of the twisted quantum affine Lie superalgebra $\mathcal{U}_{q}[\mathrm{osp}(2|2)^{(2)}]$ is presented in detail. The eigenvalues and eigenvectors of the row-to-row transfer matrix are calculated and the corresponding Bethe Ansatz equations are obtained and analyzed numerically.