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Introduction E9F5FC
Questions FFFFC0
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- What notion of adjacency enters into the Cartan matrix?
- What requires root systems to be positive definite?
- How does that relate to polar decomposition?
- Why are root systems not part of the unitary matrix?
Root systems describe the Lie algebra, which explains the "flat part" of the Lie group, thus the positive definite component in the polar decomposition.
The Cartan matrix expresses the amount of slack in the root system, so that you can move from the tip of one vector by X steps with another vector and stay within the root system.