标题： 关于有限维双代数的霍普夫包 显示英文标题

标题： On the Hopf envelope of finite-dimensional bialgebras

摘要： Hopf包络代数是一个由给定的双代数生成的自由Hopf代数。Hopf包络代数的存在性以及余自由Hopf代数的存在性是Hopf代数理论中的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有限维双代数的Hopf包络代数和余自由Hopf代数的实现方式，分别作为该双代数的一个特定商和子双代数。我们的构造也可以扩展到无限维情况，条件是双代数满足额外的条件，例如作为代数是左Artinian或者存在一个$n$反极子，后者是一个由此引入的概念。 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.