Nonassociative rings and algebras Definition

Generalized Cartan matrix

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Definition

Let $n \ge 1$ and let $A = (a_{ij})_{1 \le i, j \le n}$ be an $n \times n$ matrix with entries in $\Z$ . The matrix $A$ is a generalized Cartan matrix if:

$a_{ii} = 2$ for every $i$ ;
$a_{ij} \le 0$ for all $i \neq j$ ;
$a_{ij} = 0$ if and only if $a_{ji} = 0$ .

Remarks

Example: $A = \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}$ has diagonal entries $2$ , off-diagonal entries $-2 \le 0$ , and no zero entries, so it is a generalized Cartan matrix.

Non-example: $\begin{pmatrix} 2 & -1 \\ 0 & 2 \end{pmatrix}$ satisfies the first two clauses, but $a_{21} = 0$ while $a_{12} = -1 \neq 0$ , so it fails clause 3.

A generalized Cartan matrix $A$ is symmetrizable if there is an invertible diagonal matrix $D$ with positive entries such that $DA$ is symmetric. It is indecomposable if no reordering of the indices brings it into block diagonal form with two or more blocks. The Cartan matrices of finite-dimensional semisimple complex Lie algebras are exactly the generalized Cartan matrices whose principal minors are all positive.

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Matrix concept
Generalized Cartan matrix definition

Tag	`9G64Z4R395`
URI	`m2://9G64Z4R395`
Kind	definition
Writing	W0 stub
MSC	17B67, 17B22
Defines	Generalized Cartan matrix
Uses	Matrix
Changed	3 November 2026

Generalized Cartan matrix

Definition

Remarks

Reading path

Further reading

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