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

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Realization of a generalized cartan matrix

Definition

A minimal realization of a GCM $A$ of rank $r$ is a complex vector space $h$ with indexed linearly independent families $h_{1}, \dots, h_{n} \in h$ and $α_{1}, \dots, α_{n} \in h^{*}$ such that $α_{j} (h_{i}) = a_{i j}$ and $\dim h = 2 n - r$ . An isomorphism $T$ of realizations satisfies $T (h_{i}) = h_{i}^{'}$ and $α_{j}^{'} T = α_{j}$ for every index.

The matrix convention is Generalized cartan matrix. Existence and uniqueness up to isomorphism are supplied by Minimal realizations exist and are unique up to isomorphism ↗. Uniqueness of the isomorphism is not asserted.

Sources

Source comparison: Kleshchev, Definition 1.2.2 and Proposition 1.2.4, pp.10–12.

Depends on

Generalized cartan matrix

Used by

Kac Moody root lattice height and positive cone Definition
Simple reflections and the kac moody weyl group Definition
Minimal realizations exist and are unique up to isomorphism Proposition

Dependency tree · one level

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Sources

Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Definition 1.2.2 and Proposition 1.2.4, pp.10–12 (standard reference, not scraped)