2025/11/21 by Atabey Kaygun, Kaygun, Atabey
Mathematics · #16E40 #16S80 #17A32 #17B20 #17B56 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2511.17169
openalex publication_date 2025/11/21 · openalex created_date 2025/11/25 · openalex updated_date 2026/07/28
We provide a unified geometric realization of the classical deformation complexes. We construct GL-equivariant bilinear incidence varieties whose diagonal slices recover the varieties of associative, commutative, Leibniz, and Lie algebra structures on a finite-dimensional vector space. We prove that the fiber of the incidence map at a given algebra law is canonically isomorphic to the space of 2-cocycles in the corresponding cohomology theory (Hochschild, Harrison, Leibniz, or Chevalley--Eilenberg). Furthermore, we introduce invariant bilinear forms to define open strata of separable and semisimple algebras, and demonstrate that these strata consist of open GL-orbits, establishing the rigidity of generic points in the coarse moduli spaces for all four geometries.