On the Dynamical Yang-Baxter Equation

This talk is inspired by two previous ICM talks, by V. Drinfeld (1986), and G.Felder (1994). Namely, one of the main ideas of Drinfeld’s talk is that the quantum Yang-Baxter equation, which is an important equation arising in quantum field theory and statistical mechanics, is best understood within the framework of Hopf algebras, or quantum groups. On the other hand, in Felder’s talk, it is explained that another important equation of mathematical physics, the star-triangle relation, may (and should) be viewed as a generalization of the quantum Yang-Baxter equation, in which solutions depend on additional “dynamical” parameters. It is also explained there that to a solution of the quantum dynamical Yang-Baxter equation one may associate a kind of quantum group. These ideas gave rise to a vibrant new branch of “quantum algebra”, which may be called the theory of dynamical quantum groups. My goal in this talk is to give a bird’s eye review of this theory and its applications. The quantum dynamical Yang-Baxter equation (QDYBE) is an equation with respect to a function R : h∗ → Endh(V ⊗ V ), where h is a commutative finite dimensional Lie algebra, and V is a semisimple h-module. It reads R 12(λ− h3)R13(λ)R23(λ− h1) = R23(λ)R13(λ− h2)R12(λ) on V ⊗ V ⊗ V , where for instance R(λ − h) is defined by the formula R(λ − h3)(v1 ⊗ v2 ⊗ v3) := (