## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 21, Number 5 (2014), 859-872.

### Division algebras in linear Gr-categories

Hua-Lin Huang, Fred Van Oystaeyen, and Yinhuo Zhang

#### Abstract

We study division algebras in an arbitrary linear Gr-category, i.e., a category of finite-dimensional vector spaces graded by a group with associativity constraint given by a 3-cocycle. When the 3-cocycle is non-coboundary, this provides some interesting classes of nonassociative division algebras. In particular, when we work on Gr-categories over the field of real numbers, some quasi-associative version of the quaternions and octonions appear.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 21, Number 5 (2014), 859-872.

**Dates**

First available in Project Euclid: 1 January 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1420071858

**Digital Object Identifier**

doi:10.36045/bbms/1420071858

**Mathematical Reviews number (MathSciNet)**

MR3298482

**Zentralblatt MATH identifier**

06408731

**Subjects**

Primary: 17A35: Division algebras 20J06: Cohomology of groups 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23]

**Keywords**

division algebra group cohomology Gr-category

#### Citation

Huang, Hua-Lin; Van Oystaeyen, Fred; Zhang, Yinhuo. Division algebras in linear Gr-categories. Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 5, 859--872. doi:10.36045/bbms/1420071858. https://projecteuclid.org/euclid.bbms/1420071858