Classification of Real Solvable Lie Algebras Whose Simply Connected Lie Groups Have Only Zero or Maximal Dimensional Coadjoint Orbits

2015/04/17 by Vu, Le Anh, Van Hieu, Ha, Tuan, Nguyen Anh +2
#22E60 #FOS: Mathematics #Primary 17B #Rings and Algebras (math.RA) #Secondary 20G05

paper · doi:10.48550/arxiv.1504.04431

Abstract

In this paper we study a special subclass of real solvable Lie algebras having small dimensional or small codimensional derived ideal. It is well-known that the derived ideal of any Heisenberg Lie algebra is 1-dimensional and the derived ideal of the 4-dimensional real Diamond algebra is 1-codimensional. Moreover, all the coadjoint orbits of any Heisenberg Lie group as well as 4-dimensional real Diamond group are orbits of dimension zero or maximal dimension. In general, a (finite dimensional) real solvable Lie group is called an MD-group if its coadjoint orbits are zero-dimensional or maximal dimensional. The Lie algebra of an MD-group is called an MD-algebra and the class of all MD-algebras is called MD-class. Simulating the mentioned above characteristic of Heisenberg Lie algebras and 4-dimensional real Diamond algebra, we give a complete classification of MD-algebras having 1-dimensional or 1-codimensional derived ideals.

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