Hopf-Galois extensions and smash products
Let K be a field, let H be a commutative and cocommutative Hopf algebra which is finite dimensional over K, and let \(H^*\) denote the dual Hopf algebra. If A and B are algebras over K, such that A is H-Galois and B is \(H^*\)-Galois; one may form a smash product A{\#}B, and A{\#}B is an \(H\otimes H^*\)-Galois algebra over K. Every \(H\otimes H^*\)-Galois algebra over K can be constructed from a unit in \(H\otimes H^*\otimes H\otimes H^*\), which is a 2-cocycle in the Harrison cohomology for \(H\otimes H^*\). The author describes a normalization for such 2- cocycles; and he associates with a normal 2-cocycle, a homomorphism \(\sigma\) of the bialgebra H into itself. The \(H\otimes H^*\)-Galois algebra corresponding to the 2-cocycle is a smash product if, and only if, this map \(\sigma\) is an isomorphism. The author has made effective use of Harrison and Amitsur cohomology to prove and generalize results which were previously derivable in the language of group schemes.
commutative and cocommutative Hopf algebra
scientific article; zbMATH DE number 3987491
scientific article