Modal Algebra Over Generalized Taft Algebras

Hopf algebra was presented by Hopf when he studied topological properties of Lie group in the 1940s. Hopf algebra is an algebra system with algebra structure and coalgebra structure. Generalized Taft algebra is an important class of nonsemisimple noncommutative and noncocomrmitative Hopf algebra. In the study of the Hopf algebra theory, it has a better radiation effect.Prior to this, many scholars have explored semiprimality of smash product of semisimple Hopf algebra and module algebra. However, less is known when the Hopf algebras are non-semisimple noncommutative and noncocommutative. In the previous studies, this paper mainly studies the smash product when generalized Taft algebra act on module algebra. And get some sufficient and necessary conditions of smash product which is primality and semiprimeness. At the same time, it is confirmed that when the module algebra on the generalized Taft algebra is a field, the dimension formula of the field in its invariant subfield and the decomposition structure of the smash product of generalized Taft algebras act on the field.This thesis is divided into three chapters. In the first chapter, we recall the basic concepts and related results of Hopf algebra, modular algebra, smash product and generalized Taft, which provide the basis for the research of the following chapter. In the second chapter, the sufficient and necessary conditions for the smash product R#H of generalized Taft algebras and their modular algebras are studied. At the first, we give various H-stable subrings of associative algebra R always have nonzero invariants, and construct a counterexample. Second, We use Ore expansion to transform the smash product R#H, and the sufficient and necessary conditions of smash product are obtained. Finally, when the module algebra is a field, the dimension formula of the field in its invariant subfield, and the smash product of generalized Taft algebras act on the field must be a direct sum of n copies of the d×d matrices over the invariant subfield. Through the study found, the semiprime of smash product is determined by the action of a single shew derivation. The third chapter, the further research of the generalized Taft algebra and its modular algebra smash product of primality. Prove sufficient and necessary conditions for smash product is prime. And prove that when R is the reduced ring and domain, the sufficient and necessary conditions for the primality of smash product. Which is relate to the action of a class of elements in the generalized Taft algebra.