标题： 有限维双代数的Hopf包络 显示英文标题

标题： On the Hopf envelope of finite-dimensional bialgebras

摘要： 余代数的Hopf包络是通过给定的余代数生成的自由Hopf代数。在Hopf代数理论中，其存在性以及余自由Hopf代数的存在性是一个众所周知的事实，但它们的构造并不特别方便或友好。在本文中，我们提供了一种新的实现方式，将有限维余代数的Hopf包络和余自由Hopf代数分别作为该余代数本身的特定商和子余代数。我们的构造还可以扩展到无限维情况，前提是该余代数满足额外条件，例如作为代数是左Artinian或者接受一个$n$-反元素，后者是一个在此引入的概念。值得注意的是，我们开发的工具还允许我们对交换余代数的Hopf包络以及余交换余代数的余交换余自由Hopf代数给出新的描述。 显示英文摘要

摘要： 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 left Artinian 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.