Developing an Integrable Approach to Dynamical and Elliptic ModelS – DIADEMS

This project unifies four teams stemming from research laboratories in mathematics and theoretical physics having high world-wide scientific reputation of experts in the field of the classical and quantum integrable systems. It aims at developping new themes of research in the field of the quantum integrable systems, their underlying algebraic structures, mainly the elliptic algebras and the dynamical algebras; algebraic methods of resolutions of these systems: the representation theory of quantum groups and algebras, the methods of the Bethe Ansatz.

To this more mathematical aspects the project crosses a "transversal" physical theme - the calculation of correlation functions of quantum integrable models of the " spin chain " types which represents at the moment the major challenge and the key domain of activity for a relevant use of the quantum integrable systems in physics. This theme overlaps partially to three "tasks" described below:

Task 2. Development of the Bethe Ansatz techniques: through the generalization of " Manin's matrices"; in a calculation of correlation functions through the " form-factor "approach, and by fermionic representation; in a calculation of scalar products of Bethe vectors in a hierarchical BA.

Spot 3. Elliptic algebras: elliptic Gaudin models, partition functions of the SOS-models and 8 - vertex model with DWBC boundary conditions; Sklyanin-Odesskii-Feigin quantum and Poisson algebras.

Task 4. Dynamical Yang-Baxter algebras: correlations functions of open XXZ-model ; " dynamical Manin's matrices ", construction of new dynamical YB algebras (with non-commutative parametres and non-zero weights).