DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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Symmetrizable generalized cartan matrix

Definition

A GCM $A$ is symmetrizable if there is a diagonal matrix $D = diag (d_{1}, \dots, d_{n})$ with real $d_{i} > 0$ such that $B = D A$ is symmetric, equivalently $d_{i} a_{i j} = d_{j} a_{j i}$ for all $i, j$ . We fix one such $D$ when using a form.

The convention for GCM is Generalized cartan matrix. One may take the $d_{i}$ positive rational, or positive integer after scaling. Indeed, in each connected component of the nonzero-entry graph, fix its least index $v$ and replace $d_{i}$ by $d_{i} / d_{v}$ . Along an edge $i j$ , $d_{j} / d_{i} = a_{i j} / a_{j i}$ is positive rational. A finite path from $v$ to $i$ therefore gives rational $d_{i} / d_{v}$ ; its value is independent of the path because the original real symmetrizer supplies the same ratio. Multiplication by the finite product of denominators gives integers. Isolated vertices are assigned 1. Kleshchev writes $A = diag (ε_{i}) B$ ; our convention has $d_{i} = ε_{i}^{- 1}$ .

Sources

Source comparison: Kleshchev, §2.1, pp.26–27.

Depends on

Generalized cartan matrix

Used by

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Sources

Kleshchev, Lectures on Infinite Dimensional Lie Algebras — §2.1, pp.26–27 (standard reference, not scraped)