# Complex simple lie algebras: (de)classified

# Complex simple lie algebras: (de)classified

A Lie algebra in its most concrete form is the tangent space at the identity element of a Lie group. Using basic ideas from differential geometry, one can show that centerless, simple, compact Lie groups are determined by their Lie algebras, and vice versa. These Lie algebras in turn correspond to irreducible Euclidean root systems - certain rigid subsets of Euclidean space. One can then show that irreducible Euclidean root systems correspond to connected Dynkin diagrams, whose classification is an elementary problem in Euclidean geometry. Tracing these bijections in reverse, one lands at a classification of centerless, simple, compact Lie groups, and their associated Lie algebras. In this talk we will discuss the above process in more detail, focusing primarily on the association of a complex, simple, finite dimensional Lie algebra, to an irreducible Euclidean root system. We will also briefly discuss some connections to representation theory and knot theory.