Quantum elliptic algebras and double Yangians

الأصل · EN

Quantum universal enveloping algebras, quantum elliptic algebras and double (deformed) Yangians provide fundamental algebraic structures relevant for many integrable systems. They are described in the FRT formalism by R-matrices which are solutions of elliptic, trigonometric or rational type of the Yang--Baxter equation with spectral parameter or its generalization known as the Gervais--Neveu--Felder equation. While quantum groups and double Yangians appear as quasi-triangular Hopf algebras, this is no longer the case for elliptic algebras and the various deformations of Yangian type algebras. These structures are dealt with the framework of quasi-Hopf algebras. These algebras can be obtained from Hopf algebras through particular Drinfel'd twists satisfying the so-called shifted cocycle condition. We review these different structures and the pattern of connections between them.