双代数胚的标量扩张
Scalar extension of bicoalgebroids
摘要：
在回顾了双代数胚的定义之后，我们定义了双代数胚上的余模和模。构造了余模的幺半范畴，并定义了双代数胚上的Yetter-Drinfel'd模.证明了Yetter-Drinfel'd范畴和双代数胚一样是monoidal和pre-braided的，并且嵌入到余代数范畴的单侧中心.本文定义了双代数胚上的辫余交换余代数（BCC），并将Brzezinski和Militaru [2]以及Balint和Slachanyi [1]的标量扩张构造对偶化到双代数胚上.给出了这种结构的几个典型例子。通过确定双代数胚上的余代数范畴与其余代数范畴，我们得到了Schauenburg定理的一个弱余代数形式。最后，我们从一元的角度研究了标量扩张和辫余交换余代数。
英文摘要：
After recalling the definition of a bicoalgebroid, we define comodules and modules over a bicoalgebroid. We construct the monoidal category of comodules, and define Yetter--Drinfel'd modules over a bicoalgebroid. It is proved that the Yetter--Drinfel'd category is monoidal and pre--braided just as in the case of bialgebroids, and is embedded into the one--sided center of the comodule category. We proceed to define Braided Cocommutative Coalgebras (BCC) over a bicoalgebroid, and dualize the scalar extension construction of Brzezinski and Militaru [2] and Balint and Slachanyi [1], originally applied to bialgebras and bialgebroids, to bicoalgebroids. A few classical examples of this construction are given. Identifying the comodule category over a bicoalgebroid with the category of coalgebras of the associated comonad, we obtain a comonadic (weakened) version of Schauenburg's theorem. Finally, we take a look at the scalar extension and braided cocommutative coalgebras from a (co--)monadic point of view.
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