Conformal Galilei groups, Veronese curves, and Newton-Hooke spacetimes

TL;DR

This paper explores the structure of nonrelativistic conformal groups related to Galilei spacetime, using Veronese maps to construct their representations, and introduces Newton-Hooke spacetimes with quantized cosmological constants.

Contribution

It provides a detailed construction of conformal Galilei groups and their matrix representations using Veronese maps, and introduces Newton-Hooke spacetimes with quantized cosmological constants.

Findings

Constructed conformal Galilei groups via Veronese maps.

Derived matrix representations of these groups.

Introduced Newton-Hooke spacetimes with quantized negative cosmological constant.

Abstract

Finite-dimensional nonrelativistic conformal Lie algebras spanned by polynomial vector fields of Galilei spacetime arise if the dynamical exponent is z=2/N with N=1,2,.... Their underlying group structure and matrix representation are constructed (up to a covering) by means of the Veronese map of degree N. Suitable quotients of the conformal Galilei groups provide us with Newton-Hooke nonrelativistic spacetimes with a quantized reduced negative cosmological constant \lambda=-N.