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This talk will focus on three affine quantum groups associated to a simple Lie algebra, namely the Yangian, quantum loop algebra and elliptic quantum group. These correspond respectively to rational, trigonometric and elliptic solutions of the Yang–Baxter equation, and arise in their simplest guises as symmetries of Heisenberg XXX, XXZ and XYZ lattice models. In previous work, we showed how to construct representations of the quantum loop algebra from those of the Yangian, and vice versa, thus settling the long standing problem of relating their representations. I will explain this construction and its extension to the case of elliptic quantum groups.

This talk is based on a joint work with V. Toledano Laredo.