Simple Lie Group - Comments On The Definition

Comments On The Definition

Unfortunately there is no single standard definition of a simple Lie group. The definition given above is sometimes varied in the following ways:

• Connectedness: Usually simple Lie groups are connected by definition. This excludes discrete simple groups (these are zero-dimensional Lie groups that are simple as abstract groups) as well as disconnected orthogonal groups.
• Center: Usually simple Lie groups are allowed to have a discrete center; for example, SL(2, R) has a center of order 2, but is still counted as a simple Lie group. If the center is non-trivial (and not the whole group) then the simple Lie group is not simple as an abstract group. Some authors require that the center of a simple Lie group be finite (or trivial); the universal cover of SL(2, R) is an example of a simple Lie group with infinite center.
• R: Usually the group R of real numbers under addition (and its quotient R/Z) are not counted as simple Lie groups, even though they are connected and have a Lie algebra with no proper non-zero ideals. Occasionally authors define simple Lie groups in such a way that R is simple, though this sometimes seems to be an accident caused by overlooking this case.
• Matrix groups: Some authors restrict themselves to Lie groups that can be represented as groups of finite matrices. The metaplectic group is an example of a simple Lie group that cannot be represented in this way.
• Complex Lie algebras: The definition of a simple Lie algebra is not stable under the extension of scalars. The complexification of a complex simple Lie algebra, such as sl(n, C) is semisimple, but not simple.

The most common definition is the one above: simple Lie groups have to be connected, they are allowed to have non-trivial centers (possibly infinite), they need not be representable by finite matrices, and they must be non-abelian.