Compact Lie group

Compact Lie groups and their representation theory are important in many areas of mathematics and physics .

The circle with center 0 and radius 1 in the complex number plane is a Lie group with complex multiplication. It is compact because it is a closed and bounded subset of the plane.

definition

A compact Lie group is a Lie group which, with the underlying topology, is a compact Hausdorff space .

classification

Each simple , connected, and simply connected , compact Lie group is one of the following:

• symplectic group , ${\displaystyle Sp(n),n\geq 1}$
• special unitary group , ${\displaystyle SU(n),n\geq 3}$
• Spin group , ${\displaystyle Spin(n),n\geq 7}$
• the compact real form of one of the exceptional Lie groups . ${\displaystyle G_{2},F_{4},E_{6},E_{7},E_{8}}$

Every connected and simply connected, compact Lie group is a product of simple, connected and simply connected, compact Lie groups.

Every connected, compact Lie group has a central extension ${\displaystyle G}$

${\displaystyle 1\to A\to {\widetilde {G}}\to G\to 1}$,

where is a finite Abelian group and the product of a torus with a connected and simply connected compact Lie group . ${\displaystyle A}$ ${\displaystyle {\widetilde {G}}=T^{m}\times K}$ ${\displaystyle K}$

A compact group has finitely many connected components , so it is a finite extension of its unit component . ${\displaystyle G_{0}}$

literature

• Mark Sepanski: Compact Lie Groups , Springer Verlag 2007. ISBN 978-0387302638