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15. Cartan subalgebras
The next project is to show that the Cartan subalgebra H of a semisimple Lie algebra
L is unique up to an automorphism of L. However, it turns out to be easier to generalize
the notion of Cartan
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12. Exceptional Lie algebras and automorphisms
We have constructed four innite families of semisimple algebras:
(1)
(2)
(3)
(4)
sl(n + 1, F ) has type An
so(2n, F ) has type Dn
so(2n + 1, F ) has type Bn
sp(2n, F ) has
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11. Classification of semisimple Lie algebras
I will explain how the Cartan matrix and Dynkin diagrams describe root systems.
Then I will go through the classication using classical examples of Lie algebras following
E
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10. Weyl group and Weyl chambers
We will use the Weyl group and the geometry of Weyl chambers to prove basic properties
of root systems, such as the uniqueness of the base up to isomorphism.
Recall that the Weyl group
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9. Abstract root systems
We now attempt to reconstruct the Lie algebra based only on the information given by
the set of roots which is embedded in Euclidean space E .
9.1. Denition. Any nite subset of Euclidean space
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8.4. Root strings. This subsection is based on Erdmann and Wildon Introduction
to Lie Algebras an undergraduate textbook which is the place to look if you dont
understand something. We rst review what we have so far us
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8. Root space decomposition
Now we come to root spaces and the classication of semisimple Lie algebras using
Dynkin diagrams. My aim is to gloss over the combinatorics and emphasize the algebraic
foundations.
First a r
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6.5. Preservation of Jordan decomposition. From now on, we will assume that F is
algebraically closed of characteristic zero.
The following theorem is crucial to the next section.
Theorem 6.5.1. Suppose L gl(V ) is sem
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6. Completely reducible representations
6.1. Mo dules.
Denition 6.1.1. A representation of L is a homomorphism L gl(V ). Then V is
called a module over L. The action of x L on v V is denoted x.v .
One can describe an a
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5. Semisimple Lie algebras and the Killing form
This section follows Procesis book on Lie Groups. We will dene semisimple Lie
algebras and the Killing form and prove the following.
Theorem 5.0.9. The following are equi
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4. Jordan decomposition and Cartans criterion
Today I will explain Cartans criterion which implies that a Lie algebra is solvable. It
uses Engels Theorem (a Lie algebra is nilpotent i every element is ad-nilpotent) and
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3. Nilpotent and solvable Lie algebras
I cant nd my book. The following is from Fulton and Harris.
Denition 3.0.1. A Lie algebra is solvable if its iterated derived algebra is zero. In
other words, Dk L = 0 where DL = [
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2. Ideals and homomorphisms
2.1. Ideals.
Denition 2.1.1. An ideal in a Lie algebra L is a vector subspace I so that [LI ] I . In
other words, [ax] I for all a L, x I .
Example 2.1.2.
(1) 0 and L are always ideals in L.
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14. Isomorphism Theorem
This section contain the important theorem that two simple Lie algebras with the same
Dynkin diagram are isomorphic. The proof uses the existence of a unique maximal root
(Exercise 10.2.5). It a