explicitly. The classification is through the associated root system (for a fixed maximal torus), which in turn are classified by their Dynkin diagrams. The classification of compact, simply connected Lie groups is the same as the classification of complex semisimple Lie algebras. Indeed, if K is a simply connected compact Lie group, then the complexification of the Lie algebra of K is semisimple. Conversely, every complex semisimple Lie algebra has a compact real form isomorphic to the Lie algebra of a compact, simply connected Lie group. Maximal tori and root systems A key idea in the study of a connected compact Lie group K is the concept of a maximal torus, that is a subgroup T of K that is isomorphic to a product of several copies of and that is not contained in any larger subgroup of this type. A basic example is the case , in which case we may take to be the group of diagonal elements in . A basic result is the torus theorem which states that every element of belongs to a maximal torus and that all maximal tori are conjugate. The maximal torus in a compact group plays a role analogous to that of the Cartan subalgebra in a complex semisimple Lie algebra. In particular, once a maximal torus has been chosen, one can define a root system and a Weyl group similar to what one has for semisimple Lie algebras. These structures then play an essential role both in the classification of connected compact groups (described above) and in the representation theory of a fixed such group (described below). The root systems associated to the simple compact groups appearing in the classification of simply connected compact groups are as follows: The special unitary groups correspond to the root system The odd spin groups correspond to the root system The compact symplectic groups correspond to the root system The even spin groups correspond to the root system The exceptional compact Lie groups correspond to the five exceptional root systems G2, F4, E6, E7, or E8 Fundamental group and center It is important to know whether a connected compact Lie group is simply connected, and if not, to determine its fundamental group. For compact Lie groups, there are two basic approaches to computing the fundamental group. The first approach applies to the classical compact groups , , , and and proceeds by induction on