Smash Product Structure. Transitive Module Algebra With The Hopf-dual Galois Expansion

Smash products and Hopf-Galois extensions are important in Hopf algebra theory.In order to study a Hopf algebra, we often write it as a smash product. Hopf-Galoistheory characterizes the Hopf algebra in an elegant manner. This paper is devoted totwo aspects: one is the smash product of a semisimple Hopf algebra and its transitivemodule algebra which has a 1-dimensional ideal; the other is faithfully flat Hopf-bi-Galois extensions.Let H be a semisimple Hopf algebra over a field of characteristic 0. The properties of transitive H-module algebra A are studied. We prove that if A has a 1-dimensional ideal, then the smash product A#H is isomorphic to a full matrix algebra over some separable right coideal subalgebra N of H as algebras. It is a generation of the corresponding results of Harrison [51] and Koppinen [67]. If H is notsemisimple, a counterexample for this result is given. Moreover, for a semisimple Hopfalgebra H and its right coideal subalgebra N, there may be exist two different transitive H-module algebras such that they have a 1-dimensional ideal respectively and(?), where s = dim A = dim B. Especially, when the transitiveH-module algebra (?) the algebra of k-valued functions on a finite set (?), H is a direct sum of N₁₁-modules with samedimensions, and A#H = M_n(N₁₁), where (?).For the module algebras without 1-dimensional ideal, we prove the following resultover a fixed algebraically closed field of characteristic 0: if H is a semisimple Hopfalgebra and A a simple transitive H-module algebra, for any left ideal I of A^op, assume(A', I') is the stabilizer of (A^op, I), then smash product A#H is isomorphic to a fullmatrix algebra over A'. And A' is semisimple right H^*-module algebra. Therefore, thestudy to the smash product comes down to the semisimple algebra A'.In Chapter 4, we consider faithfully flat Hopf-bi-Galois extensions. The sufficient and necessary conditions for a Hopf-Galois extension becoming a bi-Galois extension are given. Moreover, the concrete construction is provided. Thus we generalizeSchauenburg's Hopf- bi-Galois extensions theory [99,104] over basis rings to Hopf-bi-Galois extensions of arbitrary subalgebras.