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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 integral and dominant integral weights Definition
Kac Moody root lattice height and positive cone Definition
Kac Moody Weyl vector Definition
Real coroot signs, word length and inversion sets Definition
Simple reflections and the kac moody weyl group Definition
Symmetrizable Cartan data for quantum groups Definition
The cyclic affine type A Cartan datum and its residues Definition
The formal quantum shuffle Borel and its Cartan crossed product Definition
Integrable highest weight modules for rank one gcm Example
The basic level one weight of affine sl2 Example
Affine central coroot from the transpose null ray Lemma
Finite-support regularization of quantum label-weight modules Lemma
The affine simple root alpha zero is delta minus the highest root Lemma
The opposite Borels of a symmetrizable Kac–Moody algebra are root-degreewise dual Lie bialgebras Lemma
Minimal realizations exist and are unique up to isomorphism Proposition

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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)