1. Decomposition algebras and axial algebras
- Author
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De Medts, Tom, Peacock, Simon F., Shpectorov, Sergey, and Van Couwenberghe, Michiel
- Subjects
Mathematics - Rings and Algebras ,Mathematics - Group Theory ,17A99, 20F29 - Abstract
We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras: (1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3) They admit a natural notion of homomorphisms, making them into a nice category. We exploit these facts to strengthen the connection between axial algebras and groups. In particular, we provide a definition of a universal Miyamoto group which makes this connection functorial under some mild assumptions. We illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given (permutation) group. This construction leads to a large number of examples. We also take the opportunity to fix some terminology in this rapidly expanding subject., Comment: 23 pages
- Published
- 2019
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