Nondegeneracy of simple Lie algebras of real rank 1

Work in progress with Philippe Monnier

Conn showed that compact simple Lie algebras are smoothly nondegenerate.

Weinstein showed that simple Lie algebras of real rank >= 2 are smoothly degenerate.

What about the case of real rank 1 ?

There are only 3 series of real rank 1: so(n,1), su(n,1) and sp(n,1)

(orthogonal, unitary and quaternionic)

Thm (Monnier & Z): The series su(n,1) is smoothly dengenerate.

The above theorem was not published in a journal article, but was included  in our Poisson book with JP Dufour. The proof (which is relatively simple, and included in the book) is similar to the su(1,1) = sl(2,R) case: the spiraling phenomenon (there are coadjoint orbits which have a non-trivial loop which can spiral under perturbation)

Conjecture: so(n,1) and sp(n,1) are smoothly nondegenerate

We “know” how to do it now. It will probabaly take us until February to write up the thing complete with details.

Ingredients of the proof:

1) Smooth Levi decomposition (for so(n) in so(n,1) and sp(n) in sp(n,1))

2) Formal linearization (no cohomological obstructtion)

3) Equivariant Sternberg-Chen (theorem of Belitskii-Kopanskii and the method of its proof)

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1 comment to Nondegeneracy of simple Lie algebras of real rank 1

  • admin MonsterID Icon admin

    Philippe noticed that there is an exceptional simple Lie algebra of real rank 1 that we forgot: The case F4 -20 II.

    Some informations about this algebra:

    Dimension: 52

    Real rank: 1

    Maximal compact subgroup: B4 (Spin9(R))

    Fundamental group: Order 2

    Outer automorphism group: 1

    Other names: F II

    Dimension of symmetric space: 16

    Compact symmetric space: Cayley projective plane. Quaternion-Kähler.

    Non-compact symmetric space: Hyperbolic Cayley projective plane. Quaternion-Kähler.

    Aparrently this is the only exceptional simple Lie algebra of real rank 1. Its maximal compact subgroup B4 has the same rank as itself.

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