A Duality Theorem for Hopf Quasimodule Algebras

Abstract

In this paper, we introduce and study two smash products A★H for a left H-quasimodule algebra A over a Hopf quasigroup H over a field K and B#U for a coquasi U-module algebra B over a Hopf coquasigroup U, respectively. Then, we prove our duality theorem (A★H)#H*≅A⊗(H#H*)≅A⊗Mn(K)≅Mn(A) in the setting of a Hopf quasigroup H of dimension n. As an application of our result, we consider a special case of a finite quasigroup.