Fourier transform and the ideal of group algebra on the nilpotent Engel-Lie group

Abstract

We study noncommutative Fourier transform on the nilpotent Engel-Lie group G₄ to solve some interesting problems of noncommutative analysis. In fact the finest structure of G₄ can be shown as a semi-direct product of two real vector groups. This helps us to form our idea by constructing a new larger group in order to define the Fourier transform and to obtain the Plancherel formula on G₄. Moreover we show that our methods lead us to construct several existence theorems for the invariant differential operators on G₄ and on G₄ × ℝ. Since the heat equation is invariant on G₄ × ℝ, so a fundametal solution of this equation will be obtained. Finally we establish a theorem that gives a classification of all left ideals of the noncommutative Banach algebra L¹(G₄) of G₄.