1999 Fiscal Year Final Research Report Summary
Study of monoidal categories
| Project/Area Number |
10640003
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | HIROSAKI UNIVERSITY |
|---|
Principal Investigator |
TAMBARA Daisuke Hirosaki University, Faculty of Science and Technology, associate Professor, 理工学部, 助教授 (50163712)
|
|---|
| Project Period (FY) |
1998 – 1999
|
| Keywords | tensor category / Hopt algebra / group action |
|---|
| Research Abstract |
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.
|
