Left-invariant statistical Lie groups modeled by 1-forms and their statistical Laplacian operators

We investigate left-invariant statistical structures on Lie groups whose symmetric statistical tensor is generated by a single 1-form. Indeed, the symmetric part of the connection is given by μ S = ω ⊗ Id + Id ⊗ ω , where Id is the identity operator. This condition produces a natural rank-one deformation of the Levi–Civita connection. We derive an algebraic compatibility condition between the Lie bracket and the defining 1-form that characterizes when this construction yields a statistical Lie group. Structural properties of the distinguished vector field U = ω ♯ are obtained, including the fact that U is geodesic, and the form defining the structure is determined explicitly for Lie groups of Milnor type I. Explicit formulas are established for the cubic tensor, the Tchebychev vector field, and the statistical curvature tensor. Furthermore, we explicitly compute the statistical Laplacian and provide a geometric interpretation of its intrinsic thermodynamic drift across different dimensions of hyperbolic manifolds, illustrating how statistical drift interacts with background geometry. Finally, we formally define statistical vector fields and establish that these statistical Lie groups modeled by 1-forms exhibit extreme geometric rigidity. Specifically, the resulting almost-Abelian structures admit no non-trivial left-invariant Killing vector fields. We prove that this rigidity' phenomenon is entirely independent of the choice of the left-invariant Riemannian metric, arising purely from the underlying algebraic constraints. We also establish a Bochner-type identity for the statistical Laplacian and a fundamental commutation theorem showing that these continuous symmetries commute with the statistical Laplacian, thereby preserving its eigenspaces and ensuring the symmetry of the statistical heat flow.