What is E_{8}?There actually are 4 different but related things called E_{8}.E_{8} 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: G_{2}, F_{4}, E_{6}, E_{7}, and E_{8}. The E_{8} root system consists of all vectors (called roots) (a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{7},a_{8}) 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). E_{8} has 240 roots. The 8 refers to the fact that there are 8 coordinates. See a picture of the E_{8} root system. Secondly E_{8} 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 unique even, unimodular lattice in 8 dimensions. This latter property makes it important in string theory. Next E_{8} 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. E_{8} is a 248-dimensional Lie algebra. Start with the 8 coordinates above, and add a coordinate for each of the 240 roots of the E_{8} 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 E_{8}. 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 E_{8}. This complex group has (complex) dimension 248. The E_{8} Lie algebra and group were studied by Elie Cartan in 1894. Finally E_{8} is one of three real forms of the the complex Lie group E_{8}. 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 E_{8}. Geometric description of the split real form of E_{8}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 X^{2}=-I. The set of all such matrices V_{0} 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 V_{0}, to anyone keeping score). Finally E_{8} is a group of symmetries of V. |