有限群扭等变K-理论的乘法结构
Multiplicative Structures on the Twisted Equivariant K-theory of Finite Groups
摘要：
设$K$是有限群，$G$是作用于$K$的有限群.在本文中，我们研究两个不同的，但密切相关的主题：一方面，我们分类所有可能的乘法和结合结构，人们可以赋予扭曲的$G$ -等变K-理论的$K$，另一方面，我们分类所有可能的monoidal结构，人们可以赋予扭曲和$G$ -等变丛范畴$K$。我们实现这种分类的相关信息编码的子二重复形双酒吧决议的半直积$K \rtimes G$的上链;我们使用已知的计算的上同调的$K$，$G$和$K \rtimes G$，以产生具体的例子，我们的分类。在$K=G$和$G$通过共轭作用的情况下，乘法映射$G \rtimes G \to G$是群的同态，并且我们定义了在同调水平上实现该映射的shuffle同态。我们表明，定义扭曲Drinfeld双的范畴信息可以实现为应用于$G$的任何3-上循环的shuffle同态的对偶。我们使用拉回的乘法映射上同调分类的可能的环结构的Grothendieck环的扭曲Drinfeld双表示可能有，我们包括具体的例子，这一过程。
英文摘要：
Let $K$ be a finite group and let $G$ be a finite group acting on $K$ by automorphisms. In this paper we study two different but intimately related subjects: on the one side we classify all possible multiplicative and associative structures with which one can endow the twisted $G$-equivariant K-theory of $K$, and on the other, we classify all possible monoidal structures with which one can endow the category of twisted and $G$-equivariant bundles over $K$. We achieve this classification by encoding the relevant information in the cochains of a sub double complex of the double bar resolution associated to the semi-direct product $K \rtimes G$; we use known calculations of the cohomology of $K$, $G$ and $K \rtimes G$ to produce concrete examples of our classification. In the case in which $K=G$ and $G$ acts by conjugation, the multiplication map $G \rtimes G \to G$ is a homomorphism of groups and we define a shuffle homomorphism which realizes this map at the homological level. We show that the categorical information that defines the Twisted Drinfeld Double can be realized as the dual of the shuffle homomorphism applied to any 3-cocycle of $G$. We use the pullback of the multiplication map in cohomology to classify the possible ring structures that the Grothendieck ring of representations of the Twisted Drinfeld Double may have, and we include concrete examples of this procedure.
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