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The centroid of Lie algebra is a basic concept and a necessary tool for studying the structure of Lie algebraic structure. The extended Heisenberg algebra is an important class of solvable Lie algebras. In any Lie algebra, the anti symmetry of Lie operations is an important property of Lie algebra. This article investigates the centroids and structures of 2 n + 2 dimensional extended Heisenberg algebras, where all invertible elements form a group and all elements form a ring. Then, its main research results are extended to infinite dimensional extended Heisenberg algebras. [ABSTRACT FROM AUTHOR]

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 A u t (g) = G 3 G 1 G 2 G 3 G 4 G 7 G 8 G 5 , or A u t (g) = G 3 G 1 G 2 G 3 G 4 G 7 G 8 G 5 G 6 , or A u t (g) = G 3 G 1 G 2 G 3 G 4 G 7 G 8 G 5 G 3 , where G i is a commutative subgroup of A u t (g) . We give some subgroups of g's automorphism group and systematically studied the properties of these subgroups. [ABSTRACT FROM AUTHOR]