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The exceptional cone and the Leech lattice. (English) Zbl 0863.11027

Cayley’s octonion algebra gives rise to a 27-dimensional real vector space \(J\) (the hermitian octonionic matrices of order 3) and a cubic (determinant) form \(d\) on \(J\). An integral lattice \(L\) on \(J\) is formed by the matrices with entries from Coxeter’s ring of integral octonions. It is known [the second author, Invent. Math. 124, 263-279 (1996; Zbl 0846.20049)] that \(\operatorname{Aut}(L,d)\) has two orbits on the positive definite matrices \(P\) in \(L\) with \(d(P)=1\). The authors now choose such a “polarization” \(P\) from the orbit not containing the identity matrix. They find that the corresponding Jordan root system has 819 elements, and they study the relationship between \((L,d,P)\), the unique even lattice of determinant 3 and minimum norm 4 in 26-dimensional Euclidean space (this has 144 times 819 minimal vectors), and the Leech lattice (240 times 819 minimal vectors). The paper concludes with an open question concerning the possible occurrence of the structure \((L,d,P)\) in algebraic geometry.

MSC:

11E76 Forms of degree higher than two
11H31 Lattice packing and covering (number-theoretic aspects)
14L35 Classical groups (algebro-geometric aspects)
17C40 Exceptional Jordan structures

Citations:

Zbl 0846.20049
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