On Jordan algebras that are factors of Matsuo algebras

Yıl 2024, Early Access, 1 - 21

Ilya Gorshkov Andrey Mamontov Alexey Staroletov

https://doi.org/10.24330/ieja.1488467

Öz

We describe all finite connected 3-transposition groups whose Matsuo algebras have nontrivial factors that are Jordan algebras. As a corollary, we show that if $\mathbb{F}$ is a field of characteristic 0, then there exist
infinitely many primitive axial algebras of Jordan type $\frac{1}{2}$ over $\mathbb{F}$ that are not factors of Matsuo algebras. As an example, we prove this for an exceptional Jordan algebra over~$\mathbb{F}$.

Anahtar Kelimeler

Axial algebra, Matsuo algebra, Jordan algebra, 3-transposition group

Kaynakça

• H. Cuypers and J. I. Hall, The $3$-transposition groups with trivial center, J. Algebra, 178(1) (1995), 149-193.
• T. De Medts and F. Rehren, Jordan algebras and $3$-transposition groups, J. Algebra, 478 (2017), 318-340.
• T. De Medts, L. Rowen and Y. Segev, Primitive $4$-generated axial algebras of Jordan type, Proc. Amer. Math. Soc., 152(2) (2024), 537-551.
• B. Fischer, A characterization of the symmetric groups on $4$ and $5$ letters, J. Algebra, 3 (1966), 88-98.
• B. Fischer, Finite groups generated by $3$-transpositions. I., Invent. Math., 13 (1971), 232-246.
• A. Galt, V. Joshi, A. Mamontov, S. Shpectorov and A. Staroletov, Double axes and subalgebras of Monster type in Matsuo algebras, Comm. Algebra, 49(10) (2021), 4208-4248.
• The GAP Group, GAP-Groups, Algorithms and Programming, Version 4.12.2, 2022. (https://www.gap-system.org).
• I. Gorshkov and A. Staroletov, On primitive $3$-generated axial algebras of Jordan type, J. Algebra, 563 (2020), 74-99.
• R. L. Griess, Jr., The friendly giant, Invent. Math., 69(1) (1982), 1-102.
• J. I. Hall, The general theory of $3$-transposition groups, Math. Proc. Cambridge Philos. Soc., 114(2) (1993), 269-294.
• J. I. Hall, F. Rehren and S. Shpectorov, Universal axial algebras and a theorem of Sakuma, J. Algebra, 421 (2015), 394-424.
• J. I. Hall, F. Rehren and S. Shpectorov, Primitive axial algebras of Jordan type, J. Algebra, 437 (2015), 79-115.
• J. I. Hall, Y. Segev and S. Shpectorov, Miyamoto involutions in axial algebras of Jordan type half, Israel J. Math., 223(1) (2018), 261-308.
• J. I. Hall and S. Shpectorov, The spectra of finite $3$-transposition groups, Arab. J. Math., 10(3) (2021), 611-638.
• J. I. Hall and L. H. Soicher, Presentations of some $3$-transposition groups, Comm. Algebra, 23(7) (1995), 2517-2559.
• A. A. Ivanov, The Monster Group and Majorana Involutions, Cambridge Tracts in Math., 176, Cambridge University Press, 2009.
• N. Jacobson, Some groups of transformations defined by Jordan algebras. II. Groups of type $F_4$, J. Reine Angew. Math., 204 (1960), 74-98.
• S. M. S. Khasraw, J. Mclnroy and S. Shpectorov, On the structure of axial algebras, Trans. Amer. Math. Soc., 373(3) (2020), 2135-2156.
• A. Mamontov and A. Staroletov, Axial algebras of Monster type $(2\eta,\eta)$ for $D$ diagrams. I, arXiv:2212.14608.
• A. Matsuo, $3$-transposition groups of symplectic type and vertex operator algebras, J. Math. Soc. Japan, 57(3) (2005), 639-649.
• K. McCrimmon, A Taste of Jordan Algebras, Universitext, Springer-Verlag, New York, 2004.
• J. Mclnroy and S. Shpectorov, Axial algebras of Jordan and Monster type, arXiv:2209.08043.
• B. H. Neumann, Groups whose elements have bounded orders, J. London Math. Soc., 12(3) (1937), 195-198.
• R. A. Wilson, The Finite Simple Groups, Grad. Texts in Math., 251, Springer-Verlag London, Ltd., London, 2009.
• T. Yabe, Jordan Matsuo algebras over fields of characteristic $3$, J. Algebra, 513 (2018), 91-98.
• F. Zara, A first step toward the classification of Fischer groups, Geom. Dedicata, 25(1-3) (1988), 503-512.