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Decomposition algebras and axial algebras | | Tom De Medts
; Simon F. Peacock
; Sergey Shpectorov
; Michiel Van Couwenberghe
; | | Date: |
9 May 2019 | | 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. | | Source: | arXiv, 1905.3481 | | Services: | Forum | Review | PDF | Favorites | |
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