Griess, Robert L. jun. A Moufang loop, the exceptional Jordan algebra, and a cubic form in 27 variables. (English) Zbl 0718.17028 J. Algebra 131, No. 1, 281-293 (1990). Let J be the exceptional 27-dimensional Jordan algebra over \({\mathbb{C}}\). Its automorphism group \(F_ 4({\mathbb{C}})\) contains a self-centralizing elementary abelian subgroup A of order 27. As an A-module, J decomposes into a direct sum of 1-dimensional spaces \(J_ x\), \(x\in A^{\wedge}=Hom(A,C^*)\). It was known that there are a basis of J of the form \(e_ x\in J_ x\), for \(x\in A^{\wedge}\), and a function g: \(A^{\wedge}\times A^{\wedge}\to {\mathbb{F}}_ 3\) such that \(e_ xe_ y=(-2)^{c(x,y)}\omega^{g(x,y)}e_{xy}\), where \(c(x,y)=0\) if x and y are linearly dependent and \(c(x,y)=1\) otherwise, \(\omega^ 3=1\). The elements \(e_ x\), for \(x\in A^{\wedge}\) generate the infinite commutative loop L. The loop L has as quotient a Moufang loop M of order 81 and exponent 3. The author constructs a polynomial of the third degree as a function of g(x,y). Using g and the loop M the exceptional algebra J can be constructed. The author also gets a new construction of the cubic form in 27 variables whose group is \(3E_ 6({\mathbb{C}})\). The loops associated with the exceptional Jordan algebra J are used to study important finite subgroups of sporadic simple groups and Lie groups. Here the loops are of odd order. One compares the situation with the loops of even order in the case of Cayley numbers. Reviewer: Yu.A.Medvedev (Novosibirsk) Cited in 2 ReviewsCited in 13 Documents MSC: 17C40 Exceptional Jordan structures 20N05 Loops, quasigroups 22E10 General properties and structure of complex Lie groups Keywords:Moufang loop; cubic form in 27 variables; exceptional Jordan algebra; finite subgroups of sporadic simple groups; Lie groups × Cite Format Result Cite Review PDF Full Text: DOI Link References: [1] Bruck, R. H., A Survey of Binary Systems (1958), Springer-Verlag: Springer-Verlag Berlin · Zbl 0141.01401 [2] Coxeter, H. S.M, Integral Cayley numbers, Duke Math. J., 13, 561-578 (1946) · Zbl 0063.01004 [3] Goodaire, E., Circle loops of radical alternative rings, Algebras Groups Geom., 4, 461-474 (1987) · Zbl 0659.17016 [4] Griess, R. L., Splitting of extensions of SL(3, 3) by the vector space \(F_3^3\), Pacific J. Math., 63, 405-409 (1976) · Zbl 0352.20009 [5] Griess, R. L., Code loops, J. Algebra, 100, 224-234 (1986) · Zbl 0589.20051 [6] Griess, R. L., Sporadic groups, code loops and nonvanishing cohomology, J. Pure Appl. Algebra, 44, 191-214 (1987) · Zbl 0611.20009 [8] Griess, R. L., Code loops and a large finite group containing triality for \(D_4\), (Atti del Convegno Internazionale di Teoria dei gruppi e Geometria Combinatoria. Atti del Convegno Internazionale di Teoria dei gruppi e Geometria Combinatoria, Firenze. Atti del Convegno Internazionale di Teoria dei gruppi e Geometria Combinatoria. Atti del Convegno Internazionale di Teoria dei gruppi e Geometria Combinatoria, Firenze, Rend. Circ. Mat. Palermo (2) Suppl. (22-23 Ottobre, 1986)), 79-98 · Zbl 0900.20169 [9] Griess, R. L., The friendly giant, Invent. Math., 69, 1-102 (1982) · Zbl 0498.20013 [10] Hall, M., The Theory of Groups (1959), Macmillan: Macmillan New York · Zbl 0084.02202 [11] Huppert, B., Endliche Gruppen I (1968), Springer-Verlag: Springer-Verlag Berlin [12] Jacobson, N., Structure and Representations of Jordan Algebras (1968), Amer. Math. Soc: Amer. Math. Soc Providence, RI · Zbl 0218.17010 [14] Racine, M., On maximal subalgebras, J. Algebra, 30, 155-180 (1974) · Zbl 0282.17009 [15] Racine, M., Maximal subalgebras of exceptional Jordan algebras, J. Algebra, 46, 12-21 (1977) · Zbl 0358.17018 [16] Sah, Chih-Han, Cohomology of split group extensions, J. Algebra, 29, 255-302 (1974) · Zbl 0277.20071 [17] Steinberg, R., Generators, relations and coverings of algebraic groups, J. Algebra, 71, 527-543 (1981) · Zbl 0468.20038 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.