Study of monoidal categories
Project/Area Number  10640003 
Research Category 
GrantinAid for Scientific Research (C).

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 Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 1999 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1998 : ¥500,000 (Direct Cost : ¥500,000)

Keywords  tensor category / Hopt algebra / group action / テンソルカテゴリー / ホップ代数 / 群作用 / 格別2群 / スマッシュ積 
Research Abstract 
1. For a tensor category C, one has a notion of a Cmodule by analogy with a module over a ring. Namely, a Cmodule 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 onetoone correspondence between Amodules with direct summands and Bmodules with direct summands . This gives a categorical interpretation of the wellknown duality in the smash product construction for Hopf algebra actions on rings. 2. Suppose a finite group G acts on a tensor category C. Ginvariant objects in C form a tensor category, which we denote by A. Let B be the semidirect 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 onetoone correspondence between Amodules with direct summands and Bmodules with direct summands. 3. Let F be a finite field. Let G be the semidirect 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.

Report
(4results)
Research Output
(9results)