2022/07/30 by Xavier Bekaert, Bekaert, Xavier, Niels Kowalzig +3 · 3 citations
Mathematics · #16S30 #16S40 #16W25 #17B66 #53C05 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2208.00266
openalex publication_date 2022/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
We extend a theorem, originally formulated by Blattner-Cohen-Montgomery for crossed products arising from Hopf algebras weakly acting on noncommutative algebras, to the realm of left Hopf algebroids. Our main motivation is an application to universal enveloping algebras of projective Lie-Rinehart algebras: for any given curved (resp. flat) connection, that is, a linear (resp. Lie-Rinehart) splitting of a Lie-Rinehart algebra extension, we provide a crossed (resp. smash) product decomposition of the associated universal enveloping algebra, and vice versa. As a geometric example, we describe the associative algebra generated by the invariant vector fields on the total space of a principal bundle as a crossed product of the algebra generated by the vertical ones and the algebra of differential operators on the base.