Skip to main content
SpringerLink
Log in

Code algebras which are axial algebras and their ℤ2-gradings

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

A code algebra AC is a non-associative commutative algebra defined via a binary linear code C. We study certain idempotents in code algebras, which we call small idempotents, that are determined by a single nonzero codeword. For a general code C, we show that small idempotents are primitive and semisimple and we calculate their fusion law. If C is a projective code generated by a conjugacy class of codewords, we show that AC is generated by small idempotents and so is, in fact, an axial algebra. Furthermore, we classify when the fusion law is ℤ2-graded. In doing so, we exhibit an infinite family of ℤ2× ℤ2-graded axial algebras—these are the first known examples of axial algebras with a non-trivial grading other than a ℤ2-grading.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Castillo-Ramirez, J. McInroy and F. Rehren, Code algebras, axial algebras and VOAs, Journal of Algebra 518 (2019), 146–176.

    Article  MathSciNet  Google Scholar 

  2. J. I. Hall, F. Rehren and S. Shpectorov, Universal axial algebras and a theorem of Sakuma, Journal of Algebra 421 (2015), 394–424.

    Article  MathSciNet  Google Scholar 

  3. J. I. Hall, F. Rehren and S. Shpectorov, Primitive axial algebras of Jordan type, Journal of Algebra 437 (2015), 79–115.

    Article  MathSciNet  Google Scholar 

  4. J. I. Hall, Y. Segev and S. Shpectorov, Miyamoto involutions in axial algebras of Jordan type half, Israel Journal of Mathematics 223 (2018), 261–308.

    Article  MathSciNet  Google Scholar 

  5. A. A. Ivanov, The Monster Group and Majorana Involutions, Cambridge Tracts in Mathematics, Vol. 176, Cambridge University Press, Cambridge, 2009.

  6. S. M. S. Khasraw, J. McInroy and S. Shpectorov, On the structure of axial algebras, https://doi.org/abs/1809.10132.

  7. C. H. Lam and H. Yamauchi, Binary codes and the classification of holomorphic framed vertex operator algebras, in Combinatorial Representation Theory and Related Topics, RIMS Kôkyûroku Bessatsu, Vol. B8, Research Institute for Mathematical Sciences, Kyoto, 2008, pp. 131–150.

    MathSciNet  MATH  Google Scholar 

  8. T. De Medts and M. Van Couwenberghe, Modules over axial algebras, Algebras and Representation Theorey, https://doi.org/10.1007/s10468-018-9844-y.

  9. T. De Medts and F. Rehren, Jordan algebras and 3-transposition groups, Journal of Algebra 478 (2017), 318–340.

    Google Scholar 

  10. F. Rehren, Generalised dihedral subalgebras from the Monster, Transactions of the American Mathematical Society 369 (2017), 6953–6986.

    MathSciNet  Google Scholar 

  11. F. Rehren, Linear idempotents inMatsuo algebras, Indiana UniversityMathematics Journal 65 (2016), 1713–1733.

    MathSciNet  Google Scholar 

  12. Y. Segev, Half-axes in power associative algebras, Journal of Algebra 510 (2018), 1–23.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matematicas Centro Universitario de Ciencias Exactas e Ingenierias, Universidad de Guadalajara, Blvd. Marcelino García Barragán #1421, esq Calzada Olímpica, C.P. 44430, Guadalajara, Jalisco, México

    Alonso Castillo-Ramirez

  2. School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    Justin McInroy

  3. The Heilbronn Institute for Mathematical Research, Bristol, UK

    Justin McInroy

Authors
  1. Alonso Castillo-Ramirez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Justin McInroy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alonso Castillo-Ramirez.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Castillo-Ramirez, A., McInroy, J. Code algebras which are axial algebras and their ℤ2-gradings. Isr. J. Math. 233, 401–438 (2019). https://doi.org/10.1007/s11856-019-1911-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-019-1911-5

Search

Navigation