We prove that the category of cocommutative bialgebras in any locally presentable symmetric monoidal is an [Formula: see text]-protomodular with respect to a particular class split extensions bialgebras. also obtain “partial” well-known Smith Huq condition, meaning two effective text]-equivalence relations centralize each other as soon normal subobjects associated them commute sense Huq.

An admissible Poisson algebra (or briefly, an adm-Poisson algebra) gives equivalent presentation with only one operation for a algebra. We establish bialgebra theory algebras independently and systematically, including but beyond the corresponding results on bialgebras given in [27]. Explicitly, we introduce notion of which are to Manin triples as well bialgebras. The direct correspondence betw...

By constructing explicit examples, we show that the core of a group-like element in cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra over group ring finite abelian is not always completely trivial.

The relations between the radical of crossed product R# σ H and algebra R are obtained. Using this theory, the author shows that if H is a finite-dimensional semisimple, cosemisimle, and either commutative or cocommutative Hopf algebra, then R is H-semiprime iff R is semiprime iff R# σ H is semiprime.

We discuss quantum deformation of the affine transformation algebra. It is shown that the quantum algebra has a non-cocommutative Hopf algebra structure, simple realizations and quantum tensor operators. ——————————————————————† Fellow of the Japan Society for the Promotion of Science E-mail : [email protected] ∗ E-mail : [email protected]

Endomorphisms of the measure algebra commutative hypergroups are investigated. We focus on derivations and higher order which closely related to moment function sequences rank. describe exact connection between those endomorphisms if it is considered as a module over ring continuous functions.