## Generalised dihedral subalgebras from the Monster

HTML articles powered by AMS MathViewer

- by Felix Rehren PDF
- Trans. Amer. Math. Soc.
**369**(2017), 6953-6986 Request permission

## Abstract:

The conjugacy classes of the Monster which occur in the McKay observation correspond to the isomorphism types of certain $2$-generated subalgebras of the Griess algebra. Sakuma, Ivanov and others showed that these subalgebras match the classification of vertex algebras generated by two Ising conformal vectors, or of Majorana algebras generated by two axes. In both cases, the eigenvalues $\alpha ,\beta$ parametrising the theory are fixed to $\frac {1}{4}$, $\frac {1}{32}$. We generalise these parameters and the algebras which depend on them, in particular finding the largest possible (nonassociative) axial algebras which satisfy the same key features, by working extensively with the underlying rings. The resulting algebras admit an associating symmetric bilinear form and satisfy the same $6$-transposition property as the Monster; $\frac {1}{4}$, $\frac {1}{32}$ turns out to be distinguished.## References

- Richard E. Borcherds,
*Vertex algebras, Kac-Moody algebras, and the Monster*, Proc. Nat. Acad. Sci. U.S.A.**83**(1986), no. 10, 3068–3071. MR**843307**, DOI 10.1073/pnas.83.10.3068 - J. H. Conway,
*A simple construction for the Fischer-Griess monster group*, Invent. Math.**79**(1985), no. 3, 513–540. MR**782233**, DOI 10.1007/BF01388521 - T. De Medts and F. Rehren,
*Jordan algebras and $3$-transposition groups*, 16pp, submitted. arXiv:1502.05657 - Igor Frenkel, James Lepowsky, and Arne Meurman,
*Vertex operator algebras and the Monster*, Pure and Applied Mathematics, vol. 134, Academic Press, Inc., Boston, MA, 1988. MR**996026** - The GAP group,
*GAP: Groups, Algorithms and Programming*, v4.6.2, 2013. - Robert L. Griess Jr.,
*The friendly giant*, Invent. Math.**69**(1982), no. 1, 1–102. MR**671653**, DOI 10.1007/BF01389186 - J. I. Hall, F. Rehren, and S. Shpectorov,
*Universal axial algebras and a theorem of Sakuma*, J. Algebra**421**(2015), 394–424. MR**3272388**, DOI 10.1016/j.jalgebra.2014.08.035 - J. I. Hall, F. Rehren, and S. Shpectorov,
*Primitive axial algebras of Jordan type*, J. Algebra**437**(2015), 79–115. MR**3351958**, DOI 10.1016/j.jalgebra.2015.03.026 - A. A. Ivanov,
*The Monster group and Majorana involutions*, Cambridge Tracts in Mathematics, vol. 176, Cambridge University Press, Cambridge, 2009. MR**2503090**, DOI 10.1017/CBO9780511576812 - A. A. Ivanov, D. V. Pasechnik, Á. Seress, and S. Shpectorov,
*Majorana representations of the symmetric group of degree 4*, J. Algebra**324**(2010), no. 9, 2432–2463. MR**2684148**, DOI 10.1016/j.jalgebra.2010.07.015 - Masahiko Miyamoto,
*Griess algebras and conformal vectors in vertex operator algebras*, J. Algebra**179**(1996), no. 2, 523–548. MR**1367861**, DOI 10.1006/jabr.1996.0023 - Masahiko Miyamoto,
*Vertex operator algebras generated by two conformal vectors whose $\tau$-involutions generate $S_3$*, J. Algebra**268**(2003), no. 2, 653–671. MR**2009325**, DOI 10.1016/S0021-8693(03)00096-6 - F. Rehren,
*Axial algebras*, PhD thesis, University of Birmingham, April 2015. - Shinya Sakuma,
*6-transposition property of $\tau$-involutions of vertex operator algebras*, Int. Math. Res. Not. IMRN**9**(2007), Art. ID rnm 030, 19. MR**2347298**, DOI 10.1093/imrn/rnm030 - Ákos Seress,
*Construction of 2-closed M-representations*, ISSAC 2012—Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2012, pp. 311–318. MR**3206319**, DOI 10.1145/2442829.2442874

## Additional Information

**Felix Rehren**- Affiliation: School of Mathematics, University of Birmingham, B15 2TT, United Kingdom
- MR Author ID: 1084481
- Email: rehrenf@maths.bham.ac.uk
- Received by editor(s): October 10, 2014
- Received by editor(s) in revised form: October 2, 2015
- Published electronically: March 1, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 6953-6986 - MSC (2010): Primary 20D08, 17C27, 17Dxx, 20Bxx, 13F20
- DOI: https://doi.org/10.1090/tran/6866
- MathSciNet review: 3683099