Statistical $\rho$-commutative algebras

In this article, we study Codazzi-couples of an arbitrary connection $\nabla$ with a nondegenerate 2-form $\omega$, an isomorphism $L$ on the space of derivation of $\rho$-commutative algebra $A$, which the important examples of isomorphism $L$ are almost complex and almost para-complex structures, a metric $g$ that $(g, \omega,L)$ form a compatible triple. We study a statistical structure on $\rho$-commutative algebras by the classical manner on Riemannian manifolds. Then by recalling the notions of almost (para-)Kähler $\rho$-commutative algebras, we generalized the notion of Codazzi-(para-)Kähler $\rho$-commutative algebra as a (para-)Kähler (or Fedosov) $\rho$-commutative algebra which is at the same time statistical and moreover define the holomorphic $\rho$-commutative algebras.