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arXiv · 2504.05821

On the Hopf envelope of finite-dimensional bialgebras

Alessandro Ardizzoni·Claudia Menini·Paolo Saracco
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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 $n$-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.

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Alessandro Ardizzoni, Claudia Menini, Paolo Saracco. 2025-12-18. On the Hopf envelope of finite-dimensional bialgebras. https://arxiv.org/abs/2504.05821

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