On generalized Poisson algebras: Solvability and constructions

This paper investigates nilpotent and solvable structures in generalized Poisson algebras, establishing analogues of Engel's and Lie's theorems within this context. We present several constructions of generalized Poisson algebras, including those derived from null-filiform and filiform associative commutative algebras, and explore extensions through unit adjunction and generalized Wronskian Lie algebras. Using polarization techniques, we establish fundamental equivalences between algebraic structures and characterize admissible algebras. Finally, we provide a complete classification of complex nilpotent generalized Poisson algebras up to dimension three.