SO(2) simple but not semisimple?

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In summary, there is no universally agreed upon definition of a simple Lie group. Some authors define it as a connected and non-abelian group with a non-trivial center. This appears to contradict what is stated in Mirman's book, which claims that the only simple group not semisimple is SO(2). The Wikipedia article on List of Simple Lie Groups provides more information on the various definitions and criteria for simple Lie groups.
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copernicus1
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I read in Mirman's book on group theory that SO(2) is the only simple group that is not semisimple. Can anyone explain this in terms a beginner could understand? I'm not sure how this is possible based on what I've read. Simple groups would seem to be a special case of semisimple groups.
 
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I notice the Wikipedia article on List of Simple Lie Groups (http://en.wikipedia.org/wiki/List_of_simple_Lie_groups) says

Simple Lie groups

Unfortunately, there is no generally accepted definition of a simple Lie group. In particular, it is not necessarily defined as a Lie group that is simple as an abstract group. Authors differ on whether a simple Lie group has to be connected, or on whether it is allowed to have a non-trivial center, or on whether R is a simple Lie group.

The most common definition implies that simple Lie groups must be connected, and non-abelian, but are allowed to have a non-trivial center.

- which is more than I know about the subject.
 

1. What is SO(2)?

SO(2) is a mathematical notation for the special orthogonal group in two dimensions. It consists of all 2x2 real matrices with determinant 1 and orthogonal matrices. It is often used in geometry and physics to represent rotations in two-dimensional space.

2. What does it mean for SO(2) to be simple?

A group is considered simple if it does not have any non-trivial normal subgroups. In other words, there are no subgroups within SO(2) that are closed under the group operation and have more than just the identity element in common. This makes SO(2) a fundamental building block for more complex groups.

3. How is SO(2) not semisimple?

A group is considered semisimple if it can be decomposed into simple subgroups. However, SO(2) is already a simple group, so it cannot be decomposed any further. Therefore, it is not semisimple.

4. What are some real-world applications of SO(2)?

SO(2) has many applications in physics, particularly in the study of rotational motion and quantum mechanics. It is also used in computer graphics to represent and manipulate 2D objects. In addition, it has been used in robotics for controlling the motion of two-dimensional objects.

5. Can SO(2) be extended to higher dimensions?

Yes, SO(2) can be extended to SO(n) for any positive integer n, representing rotations in n-dimensional space. However, SO(2) itself cannot be extended to include translations or other non-rotational transformations without losing its simplicity.

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