On the Hopf envelope of finite-dimensional bialgebras
Abstract
The Hopf envelope of a bialgebra is the free Hopf algebra generated by the given bialgebra. Its existence, as well as that of the cofree Hopf algebra, is a well-known fact in Hopf algebra theory, but their construction is not particularly handy or friendly. In this note, we offer a novel realisation of the Hopf envelope and of the cofree Hopf algebra of a finite-dimensional bialgebra as a particular quotient and sub-bialgebra, respectively, of the bialgebra itself. Our construction can also be extended to the infinite-dimensional case, provided that the bialgebra satisfies additional conditions, such as being right perfect as an algebra or admitting a -antipode, the latter being a notion hereby introduced. Remarkably, the machinery we develop also allows us to give a new description of the Hopf envelope of a commutative bialgebra and of the cofree cocommutative Hopf algebra of a cocommutative bialgebra.
Cite
@article{arxiv.2504.05821,
title = {On the Hopf envelope of finite-dimensional bialgebras},
author = {Alessandro Ardizzoni and Claudia Menini and Paolo Saracco},
journal= {arXiv preprint arXiv:2504.05821},
year = {2025}
}
Comments
The results previously stated only in the left Artinian case have been improved to the right perfect case, thanks to the referee's valuable comments and suggestions. Other minor updates