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What is the intuition behind simple Lie groups?

Background: Simple groups and their final-dimensional representations are one of the huge improtance topics at my university course.

My questions: Why are simple groups so special? Is it the condition of being connected, or not having nontrivial connected normal subgroup or what, that makes it interesting? And what is the difference between simple and semi-simple Lie groups in terms of representation theory?

I will appreciate any intuitive view on simple Lie groups that helps visualize this concept. Thank you.

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    $\begingroup$ I never got to Lie algebra in school, I had enough trouble with truth tables. I found that I could calculate the exponential function of an square matrix. But only certain matrices had logarithms. That led to Lie algebra. $\endgroup$ May 11 at 3:23

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Why are finite-dimensional simple Lie groups so special? Because they admit a full classification on the level of (complex) finite-dimensional simple Lie algebras by combinatorial data, e.g., root systems and Dynkin diagrams. Semisimple Lie algebras are then just direct Lie algebra sums of these simple Lie algebras.

By Weyl's Theorem, finite-dimensional linear representations of semisimple Lie algebras are completely reducible, i.e., a direct sum of irreducible representations. Again the irreducible representations of simple and semisimple Lie algebras can be classified by combinatorial data, e.g., using highest weight theory.

Nothing of this sort is true for solvable and nilpotent Lie groups. We can again pass to the level of Lie algebras, but a classification is impossible in general. Even for complex nilpotent Lie algebras, a complete classification is only known up to dimension $7$.

The representation theory of solvable Lie algebras is also very different. Over the complex numbers, every irreducible representation of a solvable Lie algebra is $1$-dimensional by Lie's Theorem, but a result as Weyl's Theorem does not hold. So there is no way to classify linear representations of nilpotent and solvable Lie algebras and Lie groups.

There are many references in the literature. The wikipedia article on Semisimple Lie Algebras has further references, as well as the article on its Representation Theory. Finally, there is the nice article on Simple Lie groups. This has applications to geometry, e.g., to Riemannian symmetric spaces.

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  • $\begingroup$ Thank you, this is fantastic! Could you pls clarify: 1) What else is "combinatorial data", does it have specific definition or source for reading? 2) What are main results (theorems) in the classification using highest weight theory?Is it the Theorem of highest weight, or do you find anything else also interesting and important? $\endgroup$ May 8 at 17:13
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    $\begingroup$ 2) There are many results, and in particular interesting applications, for example the Weyl formulas (dimension formula, character formula), so in particular the Gauß identity, and Jacobi’s triple product identity. See Section 3.4 of my lecture notes. Compare with your former question, "why I am asking" - the Weyl Dimension Formula. $\endgroup$ May 8 at 18:26
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    $\begingroup$ I just want to say big THANK YOU, your lecture notes are my favourite source so far. I will use them for preparing for finals and further studying. I like how it is complex, but still straightforward, been missing something like that at my uni courses. $\endgroup$ May 9 at 10:58
  • $\begingroup$ You are wlecome, thank you. $\endgroup$ May 9 at 11:21
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    $\begingroup$ @TerezaTizkova "combinatorial data" is a slightly loose term but refers here to the fact that a representation of a given semisimple Lie algebra can be uniquely identified by a finite list of integers. These integers represent the coefficients of the fundamental weights in the highest weight of the representation and this is pretty much the theorem of highest weight. (I should mention there are other ways we could classify the representations combinatorially). $\endgroup$
    – Callum
    May 11 at 12:55
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Another point view towards the importance of simple objets generally in mathematics, e.g. simple finite groups, simple modules (i.e. irreducible ones) and so on, is that they constitute the elementary blocks of the theory in question, and their investigations is a matter self interest in every area of mathematics.

Also, as it is always the case through the history of mathematics, the intuition beyond simple objets leads often to interesting classification (Cartan-Killing classification of simple complex Lie algebras by Cartan matrices or Dynkin diagrams), classification of simple complex Lie groups, etc. In a rudimentary analogy, those simple objets can be seen as what prime numbers and principal ideals are for number and ring theories.

Finally, those objects have many interesting properties, like having a unique invariant nondegenerate bilinear form, up to sign, called Killing form, having a completely reducible representation (Weyl's Theorem), having only trivial cohomological groups in any irreducible module (consequence of Whitehead's Lemmas). Moreover, the other important aspect is that they have plenty of applications in Geometry and Physics, e.g. classification of Riemannian symmetric spaces, study of compact Lie groups, and some simple groups as $SU(2)$ emerged in the study of symmetries in the standard model of particule physics, ... .

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