Some results in Zinbiel H-pseudoalgebras

Abstract

We define a Zinbiel [Formula: see text]-pseudoalgebra over a cocommutative Hopf algebra [Formula: see text], and characterize the annihilation algebra of Zinbiel [Formula: see text]-pseudoalgebra. The relation between Zinbiel [Formula: see text]-pseudoalgebra and its annihilation algebra is also studied. It is proved that every Zinbiel [Formula: see text]-pseudoalgebra of rank one is trivial. We study weak current Zinbiel [Formula: see text]-pseudoalgebras which are very close to current Zinbiel [Formula: see text]-pseudoalgebras and give the classification of nul-filiform and filiform weak current Zinbiel [Formula: see text]-pseudoalgebras of finite rank. Moreover, the classification of weak current Zinbiel [Formula: see text]-pseudoalgebras of rank under four is also determined.