Faithful representations of the Galilean Lie algebra in two spatial dimensions

Abstract

AbstractLet𝔊{\mathfrak{G}}be the Galilean Lie algebra in 2+1 space-times. It is known that there is no faithful representation of𝔊{\mathfrak{G}}given by derivations with homogeneous coefficients of degree 1 on the polynomial ringℝ⁢[x1,…,xn]{\mathbb{R}[x_{1},\dots,x_{n}]}forn≤3{n\leq 3}. In this paper, we consider such representations of𝔊{\mathfrak{G}}forn=4{n=4}. By classifying all collections of Galilean matrices of order 4 up to conjugation, which yields a complete classification of faithful𝔊{\mathfrak{G}}-modules of dimension 4, we show that all such faithful representations of𝔊{\mathfrak{G}}onℝ⁢[x1,…,x4]{\mathbb{R}[x_{1},\dots,x_{4}]}are classified into two types up to equivalence, each of which is parameterized byℝ3{\mathbb{R}^{3}}. As a byproduct, we show that all faithful𝔊{\mathfrak{G}}-modules of dimension 4 are indecomposable.