Integrable Models
Two dimensional integrable models have consistently attracted a great deal of research interest in recent years primarily because their study provides exact results, without recourse to perturbation theory or other approximation schemes as is the case when studying most physical problems. The main challenge when investigating a physical system is to develop exact, non-perturbative means in order to tackle physically relevant questions in the most indubitable manner possible. Integrability offers such an exact framework, and this is one of its great appeals. Beyond their physical significance, such models are also of great mathematical interest since their investigations entail the development of intriguing algebraic and geometric structures, whose study has increasingly grown over the last decades.

Factorizable Scattering-Quantum algebras
Integrable models display in general particle-like excitations, whose interactions are described by the bulk scattering S-matrix satisfying a collection of algebraic constraints, known as the Yang-Baxter equation. From the physical viewpoint, as advocated in the works of Zamolodchikov-Zamolodchikov and Faddeev-Takhtakjan, the Yang-Baxter equation describes the factorization of multi-particle scattering, a unique feature displayed by 2-d integrable systems. From a mathematical viewpoint Jimbo and Drinfeld argued independently, that the algebraic structures underlying the Yang-Baxter equation may be seen as deformations of the usual Lie algebras or their infinite dimensional extensions, the Kac-Moody algebras. Such deformed algebraic structures are endowed with a non trivial co-product and are known as quantum groups or quantum algebras; they also turn out to have deep connections with the so called braid group.

Quantum spin chain

Quantum Spin Chains-Bethe Ansatz
Many ideas in quantum integrability have their origins in statistical physics dating back in the seminal works of Bethe and Onsager, who solved the Heisenberg and Ising models respectively, as well as the celebrated solution of the XYZ model discovered by Baxter. A more recent approach on the resolution of the spectrum of 1-d statistical models is the Bethe ansatz method, an elegant algebraic technique developed mainly by the St. Petersburg group, yielding also one of the main paths towards formulating the quantum algebras. In the frame of integrable quantum spin chains the first step towards the derivation of observables is the diagonalization of the corresponding Hamiltonian, by means of the quantum inverse scattering method, which is based on the algebra underlying the model.

Hamiltonian Formulation-Lax pair
In the context of classical integrable systems Lax representation of classical dynamical evolution equations is one key ingredient together with the associated notion of classical r-matrix as suggested by Sklyanin and Semenov-Tian-Shansky. It takes the generic form of an isospectral evolution equation. The spectrum of the Lax matrix or its extension or equivalently the invariant coefficients of the characteristic determinant provide automatically candidates to realize the hierarchy of Poisson-commuting Hamiltonians required by Liouville's theorem. Existence of the classical r-matrix guarantees Poisson commutativity of these natural dynamical quantities taken as generators of the algebra of classical conserved charges.

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