Orbit-rank simplicial complexes generated by Lie algebras arising from smooth Lie group actions

In this paper, we introduce orbit-rank simplicial complexes arising from smooth Lie group actions on manifolds. The proposed construction is based on the infinitesimal geometry of orbit maps together with common intersections of stabilizer Lie algebras, yielding a simplicial complex whose vertices correspond to the non-trivial orbits of the action. This framework provides a unified description of orbit geometry and infinitesimal symmetry by combining orbit ranks with common stabilizer directions. We establish that the proposed construction is intrinsic and investigate its fundamental geometric, algebraic, and combinatorial properties. In particular, we study maximal simplices, facets, dimension bounds, and rank-induced filtrations and characterize facets through common stabilizer directions and their associated zero sets. Furthermore, we relate orbit-map differentials, stabilizer Lie algebras, tangent space decompositions, and embedded submanifolds to the simplicial structure, revealing new interactions between differential geometry and simplicial topology generated by Lie group actions. Additionally, we derive formulas for the $ f $-vector and Euler characteristic, together with their decomposition through the rank filtration, providing refined topological invariants of the proposed complexes. Several illustrative examples are presented to demonstrate the effectiveness of the construction and to verify the developed theory. The obtained results establish a new connection between differential geometry, Lie group actions, algebraic topology, and combinatorial structures, providing a systematic framework to translate infinitesimal geometric information into higher-dimensional simplicial models.