Hidden symmetries of mathematical objects

A general philosophy is that, mathematical objects have symmetry groups, and can be classified by these groups. The Galois theory is an example. Transformation groups or groupoids, linear representation theory, classification of metrics by holonomy groups, etc.,  are also instances of this philosophy.

There are objects, which a-priori have no symmetries, but still have some hidden intrinsic symmetry groups / algebras, and can still be classified by their hidden symmetries.

Examples:

– Vector fields which vanish at a point. Poincare-Birkhoff normalization amounts to the linearization of the intrinsic hidden torus symmetry group.

– Poisson structures: local normal forms -> Levi decomposition of the corresponding infinite-dimensional Lie algebra. The Levi part is the semisimple symmetry algebra.

– sub-Riemannian structures at singular points have associated solvable Lie algebras.

Now an open research question:

What is the intrinsic hidden symmetry group of a pre-symplectic structure at a singular point ? Can one use this group to normalize the structure ?

What about the other structures (e.g. singular foliations etc.) ?

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1 comment to Hidden symmetries of mathematical objects

  • hong van MonsterID Icon hong van

    Before you want to classify something you need to understand
    the notion of equivalence. And here the notion of hidden symmetry plays a role!

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