Abstract
Nonassociative commutative algebras A, generated by idempotents e whose adjoint operators ad e : A → A, given by x ↦ xe, are diagonalizable and have few eigenvalues, are of recent interest. When certain fusion (multiplication) rules between the associated eigenspaces are imposed, the structure of these algebras remains rich yet rather rigid. For example, vertex operator algebras give rise to such algebras. The connection between the Monster algebra and Monster group extends to many axial algebras which then have interesting groups of automorphisms.
Axial algebras of Jordan type η are commutative algebras generated by idempotents whose adjoint operators have a minimal polynomial dividing (x-1)x(x-η), where η ∉ {0, 1} is fixed, with well-defined and restrictive fusion rules. The case of η ≠1/2 was thoroughly analyzed by Hall, Rehren and Shpectorov in a recent paper, in which axial algebras were introduced. Here we focus on the case where η = 1/2, which is less understood and is of a different nature.
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References
R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), 3068᾿071.
N. Bourbaki, Éléments de mathématique: groupes et algèbres de Lie, Masson, Paris, 1981, Chapitres 4, 5 et 6.
H. Cuypers and J. I. Hall, The 3-transposition groups with trivial center, J.Algebra 178 (1995), 149᾿93.
J. H. Conway, A simple construction for the Fischer-Griess monster group, Invent. Math. 79 (1985), 513᾿40.
J. I. Hall, Graphs, geometry, 3-transpositions, and symplectic F2-transvection groups, Proc. London Math. Soc. (3) 58 (1989), 89᾿11.
J. I. Hall, Some 3-transposition groups with normal 2-subgroups, Proc. London Math. Soc. (3) 58 (1989), 112᾿36.
J. I. Hall, F. Rehren and S. Shpectorov, Primitive axial algebras of Jordan type, J. Algebra 437 (2015), 79᾿15.
J. I. Hall, F. Rehren and S. Shpectorov, Universal axial algebras and a theorem of Sakuma, J. Algebra 421 (2015), 394᾿24.
J. I. Hall and L. H. Soicher, Presentations of some 3-transposition groups, Comm. Algebra 23 (1995), 2517᾿559.
A. A. Ivanov, The Monster group and Majorana involutions, Cambridge Tracts in Mathematics, Vol. 176, Cambridge University Press, Cambridge, 2009.
K. McCrimmon, A taste of Jordan algebras, Universitext, Springer-Verlag, New York, 2004.
M. Miyamoto, Griess algebras and conformal vectors in vertex operator algebras, J. Algebra 179 (1996), 523᾿48.
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Hall, J.I., Segev, Y. & Shpectorov, S. Miyamoto involutions in axial algebras of Jordan type half. Isr. J. Math. 223, 261–308 (2018). https://doi.org/10.1007/s11856-017-1615-7
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DOI: https://doi.org/10.1007/s11856-017-1615-7