Study on Poisson Algebra and Automorphism of a Special Class of Solvable Lie Algebras

Abstract

We define a four-dimensional Lie algebra g in this paper and then prove that this Lie algebra is solvable but not nilpotent. Due to the fact that g is a Lie algebra, ∀x,y∈g,[x,y]=−[y,x], that is, the operation [,] has anti symmetry. Symmetry is a very important law, and antisymmetry is also a very important law. We studied the structure of Poisson algebras on g using the matrix method. We studied the necessary and sufficient conditions for the automorphism of this class of Lie algebras, and give the decomposition of its automorphism group by Aut(g)=G3G1G2G3G4G7G8G5, or Aut(g)=G3G1G2G3G4G7G8G5G6, or Aut(g)=G3G1G2G3G4G7G8G5G3, where Gi is a commutative subgroup of Aut(g). We give some subgroups of g’s automorphism group and systematically studied the properties of these subgroups.