The sextonions and $E_{7\frac 12}$

Details The sextonions and $E_{7\frac 12}$ The sextonions and $E_{7\frac 12}$

2004-02-10

arxiv.org/abs/math/0402157v2

Mathematics

Representation Theory

25 pages, to appear in Advances in Math

Scientific paper

We fill in the "hole" in the exceptional series of Lie algebras that was observed by Cvitanovic, Deligne, Cohen and deMan. More precisely, we show that the intermediate Lie algebra between $E_7$ and $E_8$ satisfies some of the decomposition and dimension formulas of the exceptional simple Lie algebras. A key role is played by the sextonions, a six dimensional algebra between the quaternions and octonions. Using the sextonions, we show simliar results hold for the rows of an expanded Freudenthal magic chart. We also obtain new interpretations of the adjoint variety of the exceptional group $G_2$.

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