On the symmetry algebras of 5-dimensional CR-manifolds

TL;DR

This paper classifies the symmetry algebras of 5-dimensional CR-hypersurfaces, showing they are either maximal with 15 dimensions for spherical cases or at most 11, with examples of the latter constructed.

Contribution

It provides a complete classification of symmetry algebra dimensions for 5D CR-hypersurfaces and constructs explicit examples with symmetry algebra dimension 11.

Findings

Maximal symmetry algebra dimension is 15 for spherical hypersurfaces.

Hypersurfaces with symmetry algebra dimension 11 are dense and spherical with Levi form signature (1,1).

Constructed examples of CR-hypersurfaces with symmetry algebra dimension 11.

Abstract

We show that for a real-analytic connected holomorphically nondegenerate 5-dimensional CR-hypersurface $M$ and its symmetry algebra $\mathfrak{s}$ one has either: (i) $\dim\mathfrak{s}=15$ and $M$ is spherical (with Levi form of signature either $(2,0)$ or $(1,1)$ everywhere), or (ii) $\dim\mathfrak{s}\le11$ where $\dim\mathfrak{s}=11$ can only occur if on a dense open subset $M$ is spherical with Levi form of signature $(1,1)$. Furthermore, we construct a series of examples of pairwise nonequivalent CR-hypersurfaces with $\dim\mathfrak{s}=11$.