Automorphism Groups Of Green Cycles On Sweedler Hopf Algebras

Hopf algebra was presented by Heinz Hopf when he studied topological properties of Lie group in the 1940s. Hopf algebra is an algebra system with algebra structure and coalgebra structure. In the 1960s, Hochscild developed and enriched Hopf algebra when he studied the application of representation of Lie group. In 1965, Milnor and Moore called the above concept Hopf algebra, which laid the foundation for the further study of Hopf algebra. In 1975, Kaplansky summarized the latest research achievements in the field of mathematics, and put forward ten famous conjectures, which further promoted the development of Hopf algebra. Taft algebra and generalized Taft algebra are two kinds of Hopf algebra, which are non-commutative and non-cocommutative. In 2012, Chen, Van Oystaeyen and Zhang determined algebra structure of Green ring (representation ring) of Taft algebra, that is generators and generating relations, and proved that the Green ring of Taft algebra is isomorphic to the quotient ring, which is equal to the polynomial algebra over Z in two variables modulo an ideal. On this basis, in 2013, Li and Zhang investigated the generators, generating relations of the Green ring of the generalized Taft Hopf algebra, and determined all nilpotent elements.In this paper, the research work of Chen, Van Oystaeyen, Zhang and Li will be continued, the automorphism group of Green ring and Green algebra of Sweedler 4-dimensional Hopf algebra will be investigated by using the structure of Green ring r{Hi) over Sweedler 4-dimensional Hopf algebra, basic knowledge of automorphisms and ring homomorphism theory. This article is composed of three parts. In the first part, we mainly recalls some basic definitions of the algebra, coalgebra, bialgebra, Hopf algebra, Taft algebra, generalized Taft algebra and Green ring of Hopf algebra. What’s more, we review some important results of the algebra structure of the Green ring of Taft algebra and generalized Taft Hopf algebra. In the second part, the images of above elements are calculated by basic elements and the generating relations up to any isomorphism. And then, the automorphism group Aut(r{H2)) of Green ring r(H2) of Sweedler 4-dimensional Hopf algebra is determined. Finally, it is proved that the automorphism group Aut(r(H2) is isomorphic to Klein group K4. In the third part, the automorphism group Aut(F(H2)) of Green algebra F(H2) of Sweedler 4-dimensional Hopf algebra over a field F, whose characteristics is not equal to 2, is investigated. Then, it is turned out that Aut(F(H2)) is isomorphic to the direct product of Z2 and a normal subgroup of Aut(F(H2)).