The sextonions and E7 1/2

Joseph M. Landsberg (1) , Laurent Manivel (2)

1 TAMU - Department of Mathematics [Texas]
2 IF - Institut Fourier

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 obtainnew interpretations of the adjoint variety of the exceptional group $G_2$.

Mots clés: exceptional Lie algebra; octonion; sextonion; Freudenthal's magic square

Domaines: Géométrie algébrique [math.AG]

Joseph M. Landsberg, Laurent Manivel. The sextonions and E7 1/2. Advances in Mathematics, 2006, 201 (1), pp.143-179. ⟨hal-00330636⟩