What is E₈?

There actually are 4 different but related things called E₈.

E₈ is first of all the largest exceptional root system, which is a set of vectors in an 8-dimensional real vector space satisfying certain properties. Root systems were classified by Wilhelm Killing in the 1890s. He found 4 infinite classes of Lie algebras, labelled A_n, B_n, C_n, and D_n, where n=1,2,3.... He also found 5 more exceptional ones: G₂, F₄, E₆, E₇, and E₈.

The E₈ root system consists of all vectors (called roots) (a₁,a₂,a₃,a₄,a₅,a₆,a₇,a₈) where all a_i are integers or all a_i are integers plus 1/2, the sum is an even integer, and sum of the squares is 2. An example with all integers is (-1,0,1,0,0,0,0,0) (there are 112 of these) and an example with half-integers is (1/2,1/2,-1/2,-1/2,-1/2,1/2,1/2,-1/2) (there are 128 of these). E₈ has 240 roots. The 8 refers to the fact that there are 8 coordinates. See a picture of the E₈ root system.

Secondly E₈ refers to the root lattice obtained by taking all sums (with integral coefficients) of the vectors in the root system. It consists of all vectors above with all a_i integers, or all a_i integers plus 1/2, and whose sum is even. The integers of squared length 2 are precisely the roots. This lattice, sometimes called the "8-dimensional diamond lattice", has a number of remarkable properties. It gives most efficient sphere-packing in 8 dimensions, and is also the uniqueeven, unimodular lattice in 8 dimensions. This latter property makes it important in string theory.

Next E₈ is a semisimple Lie algebra. A Lie algebra is a vector space, equipped with an operation called the Lie bracket. A simple example is the set of all 2 by 2 matrices. This is a 4-dimensional vector space. The Lie bracket operation is [X,Y]=XY-YX.

E₈ is a 248-dimensional Lie algebra. Start with the 8 coordinates above, and add a coordinate for each of the 240 roots of the E₈ root system. This vector space has an operation on it, called the Lie bracket: if X,Y are in the Lie algebra so is the Lie bracket [X,Y]. This is like multiplication, except that it is not commutative. Unlike the example of 2 x 2 matrices, it is very hard to write down the formula for the Lie bracket on E₈.

This is a complex Lie algebra, i.e. the coordinates are complex numbers. Associated to this Lie algebra is a (complex) Lie group, also called E₈. This complex group has (complex) dimension 248. The E₈ Lie algebra and group were studied by Elie Cartan in 1894.

Finally E₈ is one of three real forms of the the complex Lie group E₈. Each of these three real forms has real dimension 248. The group which we are referring to in this web site is the split real form of E₈.

Geometric description of the split real form of E₈

Consider 16x16 real matrices X satisfying two conditions. First of all X is a rotation matrix, i.e. its rows and columns are orthonormal. Secondly assume X²=-I. The set of all such matrices V₀ is a geometric object (a "real algebraic variety"), and it is 56-dimensional. There is a natural way to add a single circle to this to make a 57-dimensional variety V. (V=Spin(16)/SU(8), and is circle bundle over V₀, to anyone keeping score). Finally E₈ is a group of symmetries of V.