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Machine/deep learning is exploring use-cases extensions for more abstract spaces such as graphs, differential manifolds, and structured data. The most recent fruitful exchanges between geometric science of information and Lie group theory have opened new perspectives to extend machine learning on Lie groups. After the Lie group’s foundation by Sophus Lie, Felix Klein, and Henri Poincaré, based on the Wilhelm Killing study of Lie algebra, Elie Cartan achieved the classification of simple real Lie algebras and introduced affine representation of Lie groups/algebras applied systematically by Jean-Louis Koszul. In parallel, the noncommutative harmonic analysis for non-Abelian groups has been addressed with the orbit method (coadjoint representation of group) with many contributors (Jacques Dixmier, Alexander Kirillov, etc.). In physics, Valentine Bargmann, Jean-Marie Souriau, and Bertram Kostant provided the basic concepts of Symplectic Geometry to Geometric Mechanics, such as the KKS symplectic form on coadjoint orbits and the notion of Momentum map associated to the action of a Lie group. Using these tools Souriau also developed the theory of Lie Group Thermodynamics based on coadjoint representations. These set of tools could be revisited in the framework of Lie group machine learning to develop new schemes for processing structured data. Data could be also embedded in homogeneous symmetric bounded domains as Poincaré hyperbolic space (Poincaré Unit Disk) or its extensions as Siegel Spaces (Siegel Unit Disk) where a Lie Group acts transitively. These spaces have been studied and classified by Elie Cartan and Jean-Louis Koszul. This space is very useful in case of isometric embedding (e.g. in Natural Language Processing). These spaces also appeared in Information Geometry as parameters spaces of Probability densities (parameters live in sharp convex cones).