Department of Physics2024-11-0920221742-658810.1088/1742-6596/2191/1/0120112-s2.0-85124986192http://dx.doi.org/10.1088/1742-6596/2191/1/012011https://hdl.handle.net/20.500.14288/8336We introduce statistical, conjugate connection and Hessian structures on anti-commutable pre-Leibniz algebroids. Anti-commutable pre-Leibniz algebroids are special cases of local pre-Leibniz algebroids, which are still general enough to include many physically motivated algebroids such as Lie, Courant, metric and higher-Courant algebroids. They create a natural framework for generalizations of differential geometric structures on a smooth manifold. The symmetrization of the bracket on an anti-commutable pre-Leibniz algebroid satisfies a certain property depending on a choice of an equivalence class of connections which are called admissible. These admissible connections are shown to be necessary to generalize aforementioned structures on pre-Leibniz algebroids. Consequently, we prove that, provided certain conditions are met, statistical and conjugate connection structures are equivalent when defined for admissible connections. Moreover, we also show that for 'projected-torsion-free' connections, one can generalize Hessian metrics and Hessian structures. We prove that any Hessian structure yields a statistical structure, where these results are completely parallel to the ones in the manifold setting. We also prove a mild generalization of the fundamental theorem of statistical geometry. Moreover, we generalize a-connections, strongly conjugate connections and relative torsion operator, and prove some analogous results. © 2021 Published under licence by IOP Publishing Ltd.Lie algebroidGroupoidMathematicsConference proceedinghttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85124986192&doi=10.1088%2f1742-6596%2f2191%2f1%2f012011&partnerID=40&md5=bd79811540f6f8c14598691d4bda12f72200