扩展结构II：量子版本
Extending Structures II: The Quantum Version
摘要：
设A是霍普夫代数，H是余代数.我们将描述和分类到一个同构的所有霍普夫代数E通过A和H分解：即E是一个霍普夫代数，使得A是E的一个Hopf子代数，H是E中的一个子余代数，具有1_{E} \in H，乘法映射$A\otimes H \to E$是双射的.我们使用的工具是一种新的积，我们称之为统一积，在这种积的构造中，A和H由三个余代数映射连接：两个作用和一个广义上圈。一个Hopf代数作用在另一个代数上的交叉积和两个霍普夫代数的双交叉积都是统一积的特殊情况。一个霍普夫代数E通过A和H分解当且仅当E同构于A和H的一个统一积。所有这样的霍普夫代数E被分类到一个同构，该同构通过Schreier型分类定理稳定A和H。由Bichon和Kassel定义的懒惰1-上圈的余代数版本在分类定理中起着关键作用。
英文摘要：
Let A be a Hopf algebra and H a coalgebra. We shall describe and classify up to an isomorphism all Hopf algebras E that factorize through A and H: that is E is a Hopf algebra such that A is a Hopf subalgebra of E, H is a subcoalgebra in E with 1_{E} \in H and the multiplication map $A\otimes H \to E$ is bijective. The tool we use is a new product, we call it the unified product, in the construction of which A and H are connected by three coalgebra maps: two actions and a generalized cocycle. Both the crossed product of an Hopf algebra acting on an algebra and the bicrossed product of two Hopf algebras are special cases of the unified product. A Hopf algebra E factorizes through A and H if and only if E is isomorphic to a unified product of A and H. All such Hopf algebras E are classified up to an isomorphism that stabilizes A and H by a Schreier type classification theorem. A coalgebra version of lazy 1-cocycles as defined by Bichon and Kassel plays the key role in the classification theorem.
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