Comodules over weak multiplier bialgebras
Comodules over weak multiplier bialgebras
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DOI：
10.1142/s0129167x14500372
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发表时间：
2013-11
影响因子：
0.6
通讯作者：
G. Böhm
中科院分区：
文献类型：
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作者：
G. Böhm
本文是[弱乘子双代数]的续篇。阿米尔。数学。Soc。研究了域上正则弱乘子双代数上具有完全乘法的模。代替通常的（弱）双代数上的协关联作用的概念，模是通过一对相容的线性映射来定义的。证明了具有完全乘法的正则弱乘子双代数的全代数和基（co）代数都带有模结构。kahhn和Van Daele的积分[弱乘子Hopf代数的Larson-Sweedler定理，准备中]被解释为从总代数到基代数的模映射。将模的可定性推广到乘子集，我们考虑了所谓的满模的特殊类。证明了它们在基（co）代数上携带双（co）模结构，并通过基（co）代数上的（co）模张量积构成了一个一元范畴。如果正则弱乘子双代数具有一个对正，则证明了有限维满模在满模的一元范畴中具有对偶。在具有完全乘法的正则弱乘子双代数上引入Hopf模。只要有对映，就证明了Hopf模的基本定理。证明了Hopf模的范畴等价于基代数上的firm模的范畴。
This is a sequel paper of [Weak multiplier bialgebras, Trans. Amer. Math. Soc., in press] in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals [The Larson–Sweedler theorem for weak multiplier Hopf algebras, in preparation] are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to the multiplier setting, we consider the particular class of so-called full comodules. They are shown to carry bi(co)module structures over the base (co)algebra and constitute a monoidal category via the (co)module tensor product over the base (co)algebra. If a regular weak multiplier bialgebra with a full comultiplication possesses an antipode, then finite-dimensional full comodules are shown to possess duals in the monoidal category of full comodules. Hopf modules are introduced over regular weak multiplier bialgebras with a full comultiplication. Whenever there is an antipode, the Fundamental Theorem of Hopf Modules is proven. It asserts that the category of Hopf modules is equivalent to the category of firm modules over the base algebra.