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Algebra

Diagonal dominance

Lie theory

Lie algebra

Semisimple Lie algebra

Cartan subalgebra

Root-space decomposition

Root system

{c}
{title2=$\langle\beta,\alpha^\vee\rangle$}

The Cartan integer associated with two roots is
$$
\langle\beta,\alpha^\vee\rangle
=\frac{2(\beta,\alpha)}{(\alpha,\alpha)}.
$$
A crystallographic root system requires every Cartan integer to be an <integer>.

Cartan integer

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 102 / 2 / a / Solution

Cartan integer ( $⟨ β, α^{\lor} ⟩$ )

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Cartan integer (⟨β,α∨⟩)

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Cartan integer ( $⟨ β, α^{\lor} ⟩$ )

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