Classification of semi-simple Lie groups

In summary, the conversation discusses the classification of semi-simple Lie groups and the role of weight, root, and co-root lattices in this classification. It is mentioned that the fundamental representations of the group can be used to deduce these lattices, and a request is made for confirmation or a reference. It is then stated that only root systems and adjoint representations are needed for classification, and that this process is simpler for Lie algebras.
  • #1
metroplex021
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0
A while ago I heard the following two facts about semi-simple Lie groups (though I have a feeling they may have to be restricted to connected semi-simple Lie groups):

1. That semi-simple Lie groups are classified by their weight (and co-weight) and root (and co-root) lattices;
2. That all of these lattices can be deduced from the fundamental representations of the group. (So that if we have a complete set of representations we can go on and infer the group.)

Can someone confirm for me that these are indeed the case, or suggest a reference where the above are stated? (I am a physics graduate but with little pure math knowledge, so the more approachable the better.) Thanks a lot!
 
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  • #2
AFAIK you need only the root systems and the adjoint representations for the classification, not all possible representations. It's a bit easier for the Lie algebras.
 

1. What are semi-simple Lie groups?

Semi-simple Lie groups are a type of mathematical object that are used to classify and study certain types of symmetries in mathematics and physics. They are groups, which means they have a set of elements and an operation that combines them, but they also have an additional structure called a Lie algebra.

2. How are semi-simple Lie groups classified?

Semi-simple Lie groups are classified using the Cartan-Killing classification, which is based on the root system of the Lie algebra. This root system is a set of vectors that determine the structure of the Lie algebra and, in turn, the structure of the Lie group.

3. What is the importance of classifying semi-simple Lie groups?

The classification of semi-simple Lie groups is important because it allows mathematicians and physicists to understand the symmetries present in various mathematical and physical systems. These groups are used in a wide range of fields, including differential geometry, quantum mechanics, and string theory.

4. Can semi-simple Lie groups be visualized?

While it is not possible to visualize the entire structure of a semi-simple Lie group, some aspects of these groups can be visualized. For example, the root system can be represented as a diagram called a Dynkin diagram, which provides a visual representation of the symmetries present in the group.

5. Are there any real-world applications of semi-simple Lie groups?

Yes, semi-simple Lie groups have many real-world applications. They are used in physics to study the symmetries of physical systems, and in mathematics to classify and study the structure of various mathematical objects. They are also used in fields such as computer science, economics, and biology.

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