K-contact Lie groups of dimension five or greater

TL;DR

This paper proves that all K-contact Lie groups of dimension five or more are central extensions of symplectic Lie groups, using complex contact geometry and properties of the Reeb vector field.

Contribution

It establishes a structural classification of high-dimensional K-contact Lie groups through complex contact geometry techniques.

Findings

K-contact Lie groups of dimension ≥5 are central extensions of symplectic Lie groups.

The proof involves complexifying the Lie algebra and analyzing the Reeb vector field.

Diagonalizability of the Reeb vector field's adjoint action implies triviality.

Abstract

We prove that a K-contact Lie group of dimension five or greater is the central extension of a symplectic Lie group by complexifying the Lie algebra and applying a result from complex contact geometry, namely, that, if the adjoint action of the complex Reeb vector field on a complex contact Lie algebra is diagonalizable, then it is trivial.