On centerless unimodular contact Lie algebras

TL;DR

The paper proves that in unimodular contact Lie algebras, the Reeb vector's adjoint action is nilpotent unless the algebra is isomorphic to sl(2,R) or su(2), and classifies certain contact Lie algebras.

Contribution

It introduces DS-contact Lie algebras, shows their relation to known classes, and classifies five-dimensional cases, providing new insights into unimodular contact Lie algebra structures.

Findings

Reeb vector action is nilpotent in unimodular contact Lie algebras, except for sl(2,R) and su(2).

Centerless unimodular examples in DS-contact class are only sl(2,R) and su(2).

Classification of five-dimensional DS-contact Lie algebras achieved.

Abstract

We provide an elementary proof that, in a (transversely) unimodular contact Lie algebra, the adjoint action of the Reeb vector is nilpotent except when the Lie algebra is isomorphic to either $\mathfrak{sl}(2,\mathbb{R})$ or $\mathfrak{su}(2)$. We introduce a class of contact Lie algebras, called \textit{DS-contact Lie algebras}, containing all K-contact Lie algebras, and deduce from the previous result that the only centerless unimodular examples in this class are precisely $\mathfrak{sl}(2,\mathbb{R})$ and $\mathfrak{su}(2)$. This gives an alternative proof of the previously known fact that centerless unimodular Sasakian Lie algebras are isomorphic to either $\mathfrak{sl}(2,\mathbb{R})$ or $\mathfrak{su}(2)$. Some other results known to hold for Sasakian Lie algebras are generalized as well. We investigate several properties of DS-contact Lie algebras in relation to Frobenius Lie algebras, and also classify them in dimension five. Some implications for the contact Lefschetz condition are explored.