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4.3 The Chevalley involution

The symmetry between the positive and negative subalgebras 𝔫+ and 𝔫− of the Kac–Moody algebra can be rephrased formally as follows: The Kac–Moody algebra is invariant under the Chevalley involution τ, defined on the generators as
τ(h ) = − h , τ (e ) = − f , τ(f ) = − e. (4.19 ) i i i i i i
The Chevalley involution is in fact an algebra automorphism that exchanges the positive and negative sides of the algebra.

Finally, we quote the following useful theorem.

Theorem: The Kac–Moody algebra 𝔤 defined by the relations (4.10View Equation, 4.11View Equation) is simple.

The proof may be found in Kac’ book [116Jump To The Next Citation Point], page 12.

We note that invertibility and indecomposability of the Cartan matrix A are central ingredients in the proof. In particular, the theorem does not hold in the affine case, for which the Cartan matrix is degenerate and has non-trivial ideals10 (see [116Jump To The Next Citation Point] and Section 4.5).

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