Generalized Cartan matrix

A generalized Cartan matrix $𝐶 = (𝑐 𝑖 𝑗)$ is an $𝑟 \times 𝑟$ matrix such that for all $𝑖, 𝑗 \in ℕ 𝑟$ lie

$𝑐 𝑖 𝑖 = 2$
$𝑐 𝑖 𝑗 \leq 0$ if $𝑖 \neq 𝑗$ ;
$𝑐 𝑖 𝑗 = 0$ iff $𝑎 𝑗 𝑖 = 0$ .

The quiver $Γ$ associated to $𝐶$ has vertices $Γ (𝑉) = ℕ 𝑟$ and $𝑎 𝑖 𝑗 = 2 𝛿 𝑖 𝑗 - 𝑐 𝑖 𝑗$ edges between vertices $𝑖$ and $𝑗$ , thus its adjacency matrix $𝐴 = (𝑎 𝑖 𝑗)$ . A quiver which can be associated to a generalized Cartan matrix is called a Cartan quiver, hence a Cartan quiver is a quiver with

no loops
every edge having at least one edge in the opposite direction

Properties

In the context of the Mutation and numbers games, $𝐶$ transitions between populations and weights on an associated quiver.

develop | en | SemBr

Table of Contents

Generalized Cartan matrix

Properties

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