The half-infinite XXZ chain in Onsagerʼs approach

In 1992 A. Zhedanov introduced the Askey-Wilson algebra AW=AW(3) and used it to describe the Askey-Wilson polynomials. In this paper we introduce a central extension $\Delta$ of AW, obtained from AW by reinterpreting certain parameters as central elements in the algebra. We call $\Delta$ the {\it universal Askey-Wilson algebra}. We give a faithful action of the modular group ${\rm {PSL}}_2({\mathbb Z})$ on $\Delta$ as a group of automorphisms. We give a linear basis for $\Delta$. We describe the center of $\Delta$ and the 2-sided ideal $\Delta[\Delta,\Delta]\Delta$. We discuss how $\Delta$ is related to the $q$-Onsager algebra.

The Universal Askey-Wilson Algebra

Generalizations of the q-Onsager algebra are introduced and studied. In one of the simplest case and q = 1, the algebra reduces to the one proposed by Uglov–Ivanov. In the general case and q ≠ 1, an explicit algebra homomorphism associated with coideal subalgebras of quantum affine Lie algebras (simply and non-simply laced) is exhibited. Boundary (soliton non-preserving) integrable quantum Toda field theories are then considered in light of these results. For the first time, all defining relations for the underlying non-Abelian symmetry algebra are explicitly obtained. As a consequence, based on purely algebraic arguments all integrable (fixed or dynamical) boundary conditions are classified.

https://hal.archives-ouvertes.fr/hal-00398665/document

Generalized q -Onsager Algebras and Boundary Affine Toda Field Theories

Modified algebraic Bethe ansatz for XXZ chain on the segment - III - Proof

Generalizations of the q-Onsager algebra are introduced and studied. In one of the simplest case and q=1, the algebra reduces to the one proposed by Uglov-Ivanov. In the general case and $q\neq 1$, an explicit algebra homomorphism associated with coideal subalgebras of quantum affine Lie algebras (simply and non-simply laced) is exhibited. Boundary (soliton non-preserving) integrable quantum Toda field theories are then considered in light of these results. For the first time, all defining relations for the underlying non-Abelian symmetry algebra are explicitely obtained. As a consequence, based on purely algebraic arguments all integrable (fixed or dynamical) boundary conditions are classified.

Generalized q-Onsager algebras and boundary affine Toda field theories

We establish an explicit algebra isomorphism between the quantum reflection algebra for the $U_q(\hat{sl_2})$ R-matrix and a new type of current algebra. These two algebras are shown to be two realizations of a special case of tridiagonal algebras (q-Onsager).

A new current algebra and the reflection equation

We establish an explicit algebra isomorphism between the quantum reflection algebra for the $${U_q(\widehat{sl_2}) R}$$
 -matrix and a new type of current algebra. These two algebras are shown to be two realizations of a special case of tridiagonal algebras (q-Onsager).

. We deﬁne and study noncommutative crossing partitions which are a generalization of non-crossing partitions. By introducing a new cover relation on binary trees, we show that the partially ordered set of noncommutative crossing partitions is a graded lattice. This new lattice contains the Kreweras lattice, the lattice of non-crossing partitions, as a sublattice. We calculate the M¨obius function, the number of maximal chains and the number of k -chains in this new lattice by constructing an explicit EL -labeling on the lattice. By use of the EL -labeling, we recover the classical results on the Kreweras lattice. We characterize two endomorphism on the Kreweras lattice, the Kreweras complement map and the involution deﬁned by Simion and Ullman, in terms of the maps on the noncommutative crossing partitions. We also establish relations among three combinatorial objects: labeled k + 1-ary trees, k -chains in the lattice, and k -Dyck tilings.

Noncommutative crossing partitions

. A recurrence relation of the generating function of the dimer model of Fibonacci type gives a functional relation for formal power series associated to lattice paths such as a Dyck, Motzkin and Schr¨oder path. In this paper, we generalize the correspondence to the case of generalized lattice paths, k -Dyck, k -Motzkin and k -Schr¨oder paths, by modifying the recurrence relation of the dimer model. We introduce ﬁve types of generalizations of the dimer model by keeping its combinatorial structures. This allows us to express the generating functions in terms of generalized lattice paths. The weight given to a generalized lattice path involves several statistics such as size, area, peaks and valleys, and heights of horizontal steps. We enumerate the generalized lattice paths by use of the recurrence relations and the Lagrange inversion theorem.

From dimer models to generalized lattice paths

. We introduce and study the lattice of generalized partitions, called weighted partitions. This lattice possesses similar properties of the lattice of partitions. By use of the pictorial representation of a weighted partition, the total number is given by the successive Stirling transforms of the Stirling number of the second kind. We construct an explicit EL -labeling on the lattice, which implies this lattice is EL -shellable and hence shellable. We compute the M¨obius function and the characteristic polynomial by use of a pictorial representation of a maximal decreasing chain. Further, a maximal decreasing chain is shown to be bijective to a labeled rooted complete binary tree.

K. Shigechi

Papers

A new current algebra and the reflection equation

A new current algebra and the reflection equation

Noncommutative crossing partitions

From dimer models to generalized lattice paths

On the lattice of weighted partitions