Introduction to Lie groups, Lie algebras, and Yang-Baxter equation

The Yang-Baxter equation "first" appeared in theoretical physics (more exactly, in the theory of integrable models), in a paper by the Nobel laureate C. N. Yang in 1960s, and in statistical mechanics, in R. J. Baxter’s work in 1970s. However, both Yang and Baxter used the equation implicitly and the explicit formulation appeared around 1980 in works of the Leningrad school led by L.D. Faddeev. Later, it turned out that this equation plays a crucial role in knot theory, braided categories, analysis of integrable systems, quantum mechanics, quantum computing, non-commutative geometry, etc. The use of the Yang–Baxter equation led to the definition of the quantum group and Lie bialgebra by Vladimir Drinfeld in 1983-1986. In my talk I will start with an overview of Lie groups and Lie algebras. Then I will define quantum (easy) and classical (more complicated) Yang-Baxter equations. If time allows, I will define Lie bialgebras and their Lie group counterpart, the so-called Poisson-Lie groups.