Borcherds’ proof of the moonshine conjecture
نویسنده
چکیده
These CSG notes contain a condensed account of a talk by V. Nikulin in the London algebra Colloquium on 24 May 2001. None of the content is original to me: it is provided simply as a service for those who missed Nikulin’s talks. I have relied mainly on my notes from the lectures, So any errors are the product of the note-taking and are not to be attributed to the content of the lectures. 1 The monster The monster, or Fischer-Griess group,M (otherwise known as the Friendly Giant) is the largest sporadic simple group. Its order is 808017424794512875886459904961710757005754368000000000 = 246 ·320 ·59 ·76 ·112 ·133 ·17 ·19 ·23 ·29 ·31 ·41 ·47 ·59 ·71. It was discovered by Fischer and Griess in 1973 and constructed by Griess in 1982. The Monster has 194 conjugacy classes (a very small number for a simple group of this size). The smallest faithful permutation representation has degree roughly 1030 (very large). The smallest faithful matrix representation over C has degree 196883; the second smallest, 21296876. Griess constructed M as the automorphism of a commutative non-associative algebra with identity on a real vector space of dimension 196884 (on which it acts as the sum of the trivial representation and the representation of degree 196883). This algebra also has anM-invariant inner product, since the representation ofM is self-dual. This algebra is known as the Griess algebra.
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