Rigidity of 2-step Carnot groups

TL;DR

This paper investigates the rigidity properties of 2-step Carnot groups and graded 2-step nilpotent Lie algebras, providing criteria and classifications based on bi-dimensions and specific algebraic conditions.

Contribution

It establishes a dichotomy for these Lie algebras, showing they are either always of infinite type or generically rigid, with explicit criteria for pseudo H- and J-type algebras.

Findings

Lie algebra structure depends on bi-dimensions, determining rigidity or infinite type.

Explicit criteria for rigidity of pseudo H- and J-type algebras are provided.

The $J^2$-condition is related to rigidity and explored within pseudo $H$-type algebras.

Abstract

In the present paper we study the rigidity of 2-step Carnot groups, or equivalently, of graded 2-step nilpotent Lie algebras. We prove the alternative that depending on bi-dimensions of the algebra, the Lie algebra structure makes it either always of infinite type or generically rigid, and we specify the bi-dimensions for each of the choices. Explicit criteria for rigidity of pseudo $H$- and $J$-type algebras are given. In particular, we establish the relation of the so-called $J^2$-condition to rigidity, and we explore these conditions in relation to pseudo $H$-type algebras.