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

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