Study of monoidal categories

1.对于张量范畴C，类比于环上的模，有C-模的概念。也就是说，C-模是C作用于其上的线性范畴。设A是有限维半单Hopf代数，B是A的对偶Hopf代数，C和D分别是A和B的有限维表示的张量范畴。建立了A-模与B-模之间的自然一一对应关系。这给出了环上Hopf代数作用的Smash积构造中众所周知的对偶性的范畴解释。设有限群G作用在张量范畴C上，C中的G-不变对象构成一个张量范畴，记为A.设B是C和G的半直积，它是一个张量范畴，其定义方式类似于斜群环.设G在基域上的群代数是半单的。得到了直和的A-模与直和的B-模之间的一一对应关系。设F是有限域。设G是F的加群与F的乘群的半直积，C是G的表示的张量范畴。与C具有相同融合规则的半单张量范畴可称为C的变形。我们得到了几个C的变形的例子。当F是三元素域时，恰好有两个变形(不同于C本身)。当F为四元场时，存在唯一的形变。当F是八元素域时，至少有一个变形。

1. For a tensor category C, one has a notion of a C-module by analogy with a module over a ring. Namely, a C-module is a linear category on which C acts. Let A be a finite dimensional semisimple Hopf algebra and B the dual Hopf algebra of A. Let C and D be the tensor categories of finite dimensional representations of A and B, respectively. We set up a natural one-to-one correspondence between A-modules with direct summands and B-modules with direct summands . This gives a categorical interpretation of the well-known duality in the smash product construction for Hopf algebra actions on rings.2. Suppose a finite group G acts on a tensor category C. G-invariant objects in C form a tensor category, which we denote by A. Let B be the semi-direct product of C and G, which is a tensor category defined in a similar manner to a skew group ring. Assume that the group algebra of G over the base field is semisimple. We obtained a one-to-one correspondence between A-modules with direct summands and B-modules with direct summands.3. Let F be a finite field. Let G be the semi-direct product of the additive group of F and the multiplicative group of F. Let C be the tensor category of representatoins of G. A semisimple tensor category having the same fusion rule as c may be called a deformation of C. We obtained a few example of deformations of C. When F is the three element field, there are exactly two deformation (other than c itself) . When F is the four element field, there is a unique deformation. When F is the eight element field, there is at least one deformation.