Reflection $K$-matrices related to Temperley-Lieb $R$-matrices

Junction type representations of the Temperley-Lieb algebra and associated symmetries

Inspired by earlier works on representations of the Temperley-Lieb algebra we introduce a novel family of representations of the algebra. This may be seen as a generalization of the so called asymmetric twin representation. The underlying symmetry algebra is also examined and it is shown that in addition to certain obvious exact quantum symmetries non trivial quantum algebraic realizations that exactly commute with the representation also exist. Non trivial representations of the boundary Temperley-Lieb algebra as well as the related residual symmetries are also discussed. The corresponding no…

On the ${\cal{U}}_{q}[osp(1|2)]$ Temperley-Lieb model

This work concerns the boundary integrability of the ${\cal{U}}_{q}[osp(1|2)]$ Temperley-Lieb model. We constructed the solutions of the graded reflection equations in order to determine the boundary terms of the correspondig spin-1 Hamiltonian. We obtain the eigenvalue expressions as well as its associated Bethe ansatz equations by means of the coordinate Bethe ansatz. These equations provide the complete description of the spectrum of the model with diagonal integrable boundaries.

Multicolored Temperley-Lieb lattice models. The ground state

Using inversion relation, we calculate the ground state energy for the lattice integrable models, based on a recently obtained baxterization of non trivial multicolored generalization of Temperley-Lieb algebras. The simplest vertex and IRF models are analyzed and found to have a mass gap.

Tensor space representations of Temperley-Lieb algebra and generalized permutation matrices

Orthogonal projections in ${\mathbb C}^n \otimes {\mathbb C}^n$ of rank one and rank two that give rise to unitary tensor space representations of the Temperley-Lieb algebra $TL_N(Q)$ are considered. In the rank one case, a complete classification of solutions is given. In the rank two case, solutions with $Q$ varying in the ranges $[2n/3,\infty)$ and $[n/\sqrt{2},\infty)$ are constructed for $n=3k$ and $n=4k$, $k \in {\mathbb N}$, respectively.

A Bethe ansatz solution for the closed $U_{q}[sl(2)]$ Temperley-Lieb quantum spin chains

We solve the spectrum pf the closed Temperley-Lieb quantum spin chains using the coordinate Bethe ansatz. These Hamiltonians are invariante under the quantum group $U_{q}[sl(2)]$

On the ${\cal{U}}_{q}[sl(2)]$ Temperley-Lieb reflection matrices

This work concerns the boundary integrability of the spin-s ${\cal{U}}_{q}[sl(2)]$ Temperley-Lieb model. A systematic computation method is used to constructed the solutions of the boundary Yang-Baxter equations. For $s$ half-integer, a general $2s(s+1)+3/2$ free parameter solution is presented. It turns that for $s$ integer, the general solution has $2s(s+1)+1$ free parameters. Moreover, some particular solutions are discussed.

Quantum symmetry algebras of spin systems related to Temperley-Lieb R-matrices

A reducible representation of the Temperley-Lieb algebra is constructed on the tensor product of n-dimensional spaces. One obtains as a centraliser of this action a quantum algebra (a quasi-triangular Hopf algebra) U_q with a representation ring equivalent to the representation ring of the sl_2 Lie algebra. This algebra U_q is the symmetry algebra of the corresponding open spin chain.

The dilute Temperley-Lieb O($n=1$) loop model on a semi infinite strip: the sum rule

This is the second part of our study of the ground state eigenvector of the transfer matrix of the dilute Temperley-Lieb loop model with the loop weight $n=1$ on a semi infinite strip of width $L$. We focus here on the computation of the normalization (otherwise called the sum rule) $Z_L$ of the ground state eigenvector, which is also the partition function of the critical site percolation model. The normalization $Z_L$ is a symmetric polynomial in the inhomogeneities of the lattice $z_1,..,z_L$. This polynomial satisfies several recurrence relations which we solve independently in terms of Ja…

Bethe ansatz solution of the anisotropic correlated electron model associated with the Temperley-Lieb algebra

A recently proposed strongly correlated electron system associated with the Temperley-Lieb algebra is solved by means of the coordinate Bethe ansatz for periodic and closed boundary conditions.

Temperley-Lieb可积模型与融合范畴

Temperley-Lieb integrable models and fusion categories

本文通过Temperley-Lieb代数结构，研究融合范畴中可积任何子链的谱特性，证明当量子维度大于2时模型存在间隙，且在特定融合范畴中有限尺寸效应可能影响间隙分析。

Temperley-Lieb K-matrices

This work concerns to the studies of boundary integrability of the vertex models from representations of the Temperley-Lieb algebra associated with the quantum group ${\cal U}_{q}[X_{n}]$ for the affine Lie algebras $X_{n}$ = $A_{1}^{(1)}$, $B_{n}^{(1)}$, $C_{n}^{(1)}$ and $D_{n}^{(1)}$. A systematic computation method is used to constructed solutions of the boundary Yang-Baxter equations. We find a $2n^{2}+1$ free parameter solution for $A_{1}^{(1)} $ spin-$(n-1/2)$ and $ C_{n}^{(1)}$ vertex models. It turns that for $A_{1}^{(1)} $ spin-$n$, $ B_{n}^{(1)}$ and $D_{n}^{(1)}$ vertex models, the…

Bethe Ansatz solutions for Temperley-Lieb Quantum Spin Chains

We solve the spectrum of quantum spin chains based on representations of the Temperley-Lieb algebra associated with the quantum groups ${\cal U}% _{q}(X_{n})$ for $X_{n}=A_{1},$ $B_{n},$ $C_{n}$ and $D_{n}$. The tool is a modified version of the coordinate Bethe Ansatz through a suitable choice of the Bethe states which give to all models the same status relative to their diagonalization. All these models have equivalent spectra up to degeneracies and the spectra of the lower dimensional representations are contained in the higher-dimensional ones. Periodic boundary conditions, free boundary c…

关于与Temperley-Lieb代数相关的模型中的关联函数

On correlation functions in models related to the Temperley-Lieb algebra

本文研究了哈密顿量源于Temperley-Lieb代数表示的量子自旋链，提出两个关键猜想，将局部算子的平均值与XXZ链的因子化关联函数联系起来，并在黄金链、$Q$态Potts模型等具体表示中进行了验证。