Golden Poisson Geometry : from algebraic structures to Lie algebras ans Poisson manifolds

Source abstract

We introduce a unified framework for Golden Poisson Geometry, built around an operator $\varphi$ satisfying the Golden identity $\varphi^2=\varphi+I$. By developing the notions of Golden algebras, Golden Poisson algebras, and Golden Lie algebras, we describe how $\varphi$ governs new compatibility patterns between algebraic operations, Lie brackets, and Poisson tensors. Linear Golden Poisson structures naturally arise on the duals of Golden Lie algebras, revealing Poisson manifolds endowed with intrinsic Fibonacci-type symmetries. Conceptual parallels with the Poisson–Nijenhuis theory of Kosmann-Schwarzbach (1990) show how Golden operators extend classical recursion phenomena. This framework opens new directions in deformation theory and in the geometric study of polynomial identities. Our main results include the construction of an infinite Golden-Fibonacci hierarchy of compatible Poisson tensors . We establish Golden recurrence relations for hamiltonian vector fields and prove functorial properties showing that the Golden Poisson morphisms transport the entire hierarchical structure. We completely characterize the compatibility conditions between Golden structures and linear Poisson brackets, analyze the associated symplectic foliations, and develop the theory of Golden Poisson morphisms.