Some Graded Hopf-module Pseudoalgebras

In this paper, we define a G-graded Hopf module associative pseudoalgebra A for any grading group G and Hopf algebra H, and construct a smash product A#H of A with H, which is also a G-graded associative pseudoalgebra. Then we establish a Mortia context between A#H and AH, the fixed sub-pseudoalgebra under the action of H. In addition, we prove a similarity of the Schur-Weyl type duality between the group algebra k[Se(n)] of Sergeev group Se(n) and a super pseudoalgebra.