Basis of the free Gelfand–Dorfman (GD) algebra

Determine a linear basis for the free Gelfand–Dorfman (GD) algebra over a field of characteristic zero, providing an explicit set of monomials or normal forms that spans the algebra and is linearly independent.

Background

The paper situates transposed Poisson and F-manifold algebras within the broader landscape of GD-algebras, noting their connections to quadratic Lie conformal algebras and Hamiltonian operators. Despite progress on special cases (e.g., the free special GD-algebra), the general problem of constructing a basis for free GD-algebras remains unresolved.

Resolving this would clarify the structure of GD-algebras and aid in related basis problems for allied operads and varieties.