2025/04/08 by Alessandro Ardizzoni, Ardizzoni, Alessandro, Claudia Menini +3 · 2 citations
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2504.05821
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.