Submaximally symmetric CR-structures

TL;DR

This paper determines the second-highest possible symmetry dimensions for hypersurface type CR-structures with non-degenerate Levi form, distinguishing between Levi-definite and Levi-indefinite cases, and confirms known results for the special case of CR-dimension one.

Contribution

It establishes the submaximal symmetry dimensions for higher CR-structures, extending and refining previous classifications, especially differentiating between Levi-definite and Levi-indefinite cases.

Findings

Maximal symmetry dimension is n^2+4n+3.

Submaximal symmetry dimension is n^2+4 for Levi-indefinite structures.

Submaximal symmetry dimension is n^2+3 for Levi-definite structures.

Abstract

Hypersurface type CR-structures with non-degenerate Levi form on a manifold of dimension $(2n+1)$ have maximal symmetry dimension $n^2+4n+3$. We prove that the next (submaximal) possible dimension for a (local) symmetry algebra is $n^2+4$ for Levi-indefinite structures and $n^2+3$ for Levi-definite structures when $n>1$. In the exceptional case of CR-dimension $n=1$, the submaximal symmetry dimension 3 was computed by E.\,Cartan.