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Modules over axial algebras

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Abstract
We introduce axial representations and modules over axial algebras as new tools to study axial algebras. All known interesting examples of axial algebras fall into this setting, in particular the Griess algebra whose automorphism group is the Monster group. Our results become especially interesting for Matsuo algebras. We vitalize the connection between Matsuo algebras and 3-transposition groups by relating modules over Matsuo algebras with representations of 3-transposition groups. As a by-product, we define, given a Fischer space, a group that can fulfill the role of a universal 3-transposition group.
Keywords
Axial algebras, Modules, Axial representations, Matsuo algebras, 3-transposition groups, Fischer spaces

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Citation

Please use this url to cite or link to this publication:

MLA
De Medts, Tom, and Michiel Van Couwenberghe. “Modules over Axial Algebras.” ALGEBRAS AND REPRESENTATION THEORY, vol. 23, no. 1, 2020, pp. 209–27, doi:10.1007/s10468-018-9844-y.
APA
De Medts, T., & Van Couwenberghe, M. (2020). Modules over axial algebras. ALGEBRAS AND REPRESENTATION THEORY, 23(1), 209–227. https://doi.org/10.1007/s10468-018-9844-y
Chicago author-date
De Medts, Tom, and Michiel Van Couwenberghe. 2020. “Modules over Axial Algebras.” ALGEBRAS AND REPRESENTATION THEORY 23 (1): 209–27. https://doi.org/10.1007/s10468-018-9844-y.
Chicago author-date (all authors)
De Medts, Tom, and Michiel Van Couwenberghe. 2020. “Modules over Axial Algebras.” ALGEBRAS AND REPRESENTATION THEORY 23 (1): 209–227. doi:10.1007/s10468-018-9844-y.
Vancouver
1.
De Medts T, Van Couwenberghe M. Modules over axial algebras. ALGEBRAS AND REPRESENTATION THEORY. 2020;23(1):209–27.
IEEE
[1]
T. De Medts and M. Van Couwenberghe, “Modules over axial algebras,” ALGEBRAS AND REPRESENTATION THEORY, vol. 23, no. 1, pp. 209–227, 2020.
@article{8616788,
  abstract     = {We introduce axial representations and modules over axial algebras as new tools to study axial algebras. All known interesting examples of axial algebras fall into this setting, in particular the Griess algebra whose automorphism group is the Monster group. Our results become especially interesting for Matsuo algebras. We vitalize the connection between Matsuo algebras and 3-transposition groups by relating modules over Matsuo algebras with representations of 3-transposition groups. As a by-product, we define, given a Fischer space, a group that can fulfill the role of a universal 3-transposition group.},
  author       = {De Medts, Tom and Van Couwenberghe, Michiel},
  issn         = {1386-923X},
  journal      = {ALGEBRAS AND REPRESENTATION THEORY},
  keywords     = {Axial algebras,Modules,Axial representations,Matsuo algebras,3-transposition groups,Fischer spaces},
  language     = {eng},
  number       = {1},
  pages        = {209--227},
  title        = {Modules over axial algebras},
  url          = {http://dx.doi.org/10.1007/s10468-018-9844-y},
  volume       = {23},
  year         = {2020},
}

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