Polarised Lie groups contactomorphic to stratified groups

TL;DR

This paper introduces modifications of stratified Lie groups called polarised Lie groups, which are locally contactomorphic to the original groups, and characterizes when such groups are modifications based on Tanaka prolongation.

Contribution

It constructs a new class of polarised Lie groups called modifications and establishes a characterization criterion based on Tanaka prolongation for their contactomorphism to stratified groups.

Findings

Modifications of stratified groups are locally contactomorphic to the original groups.

A polarised group is a modification of a stratified group if and only if it is locally contactomorphic and the Lie algebra has finite Tanaka prolongation.

The work provides a classification framework for polarised Lie groups related to stratified groups.

Abstract

For a stratified group $G$, we construct a class of polarised Lie groups, which we call modifications of $G$, that are locally contactomorphic to it. Vice versa, we show that if a polarised group is locally contactomorphic to a stratified group $G$, whose Lie algebra has finite Tanaka prolongation, then it must be a modification of $G$.