E8 Structure visualized : all 248 dimensions!

March 19th, 2007

Because of the its uniqueness and the ability to accommodate highly symmetric structure, E8 has been a favorite among physicist as a candidate for grand - super - unification! I think I wrote a couple of papers about E8 too!

The gargentuan E8 can now be visualized a little better. Below is the picture of the root system - awesome!

Lie symmetry groups come in families. The classical groups A1, A2, A3, … B1, B2, B3, … C1, C2, C3, … and D1, D2, D3, … rise like gentle rolling hills towards the horizon. Jutting out of this mathematical landscape are the jagged peaks of the exceptional groups G2, F4, E6, E7 and, towering above them all, E8. E8 is an extraordinarily complicated group: it is the symmetries of a particular 57-dimensional object, and E8 itself is 248-dimensional!

The above is a picture of the 248-dimensional Lie algebra of E8. This Lie algebra is a complex vector space, of dimension 248. It is equipped with a Lie bracket operator: for X,Y in the Lie algebra, so is [X,Y]. There are 248 nodes in the picture, one for each basis element of the Lie algebra. Label the nodes 1,…,248, and the corresponding elements of the Lie algebra Z1,…,Z248.

Visit http://aimath.org/E8/ for even more intricate pictures and appreciate the scope of the work done!

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