Mathematics – Quantum Algebra
Scientific paper
2010-02-09
Mathematics
Quantum Algebra
Version 1: 74 pages. Version 2: We split this paper into two papers, the E7 part and the E6 part. E7 part: 44 pages, E6 part:
Scientific paper
E7 part: In this paper, we study McKay's E7 observation on the Baby Monster. By investigating so called derived c=7/10 Virasoro vectors, we show that there is a natural correspondence between dihedral subgroups of the Baby Monster and certain subalgebras of the Baby Monster vertex operator algebra which are constructed by the nodes of the affine E7 diagram. This allows us to reinterpret McKay's E7 observation via the theory of vertex operator algebras. For a class of vertex operator algebras including the Moonshine module, we will show that the product of two Miyamoto involutions associated to derived c=7/10 Virasoro vectors in certain commutant vertex operator algebras is an element of order at most 4. For the case of the Moonshine module, we obtain the Baby monster vertex operator algebra as the commutant and we can identify the group generated by these Miyamoto involutions with the Baby Monster and recover the {3,4}-transposition property of the Baby Monster in terms of vertex operator algebras. E6 part: In this paper, we study McKay's E6-observation on the largest Fischer 3-transposition group Fi24. We investigate a vertex operator algebra VF of central charge 23+1/5 on which the Fischer group Fi24 naturally acts. We show that there is a natural correspondence between dihedral subgroups of Fi24 and certain vertex operator subalgebras constructed by the nodes of the affine E6 diagram by investigating so called derived Virasoro vectors of central charge 6/7. This allows us to reinterpret McKay's E6-observation via the theory of vertex operator algebras. It is also shown that the product of two non-commuting Miyamoto involutions of sigma-type associated to derived c=6/7 Virasoro vectors is an element of order 3, under certain general hypotheses on the vertex operator algebra. For the case of VF, we identify these involutions with the 3-transpositions of the Fischer group Fi24.
Hoehn Gerald
Lam Ching Hung
Yamauchi Hiroshi
No associations
LandOfFree
"McKay's E7 observation on the Baby Monster" and "McKay's E6 observation on the largest Fischer group" does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with "McKay's E7 observation on the Baby Monster" and "McKay's E6 observation on the largest Fischer group", we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and "McKay's E7 observation on the Baby Monster" and "McKay's E6 observation on the largest Fischer group" will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-124255