E8 (mathematics)

Template:Downsize In mathematics, E₈ is the name of a root system and of several associated Lie groups and also their Lie algebras ${\mathfrak {e}}_{8}$ . These are the largest of the exceptional simple Lie groups. It is also one of the simply laced groups.

E₈ has rank 8 and dimension 248. It is simply connected and its center is the trivial subgroup. Its outer automorphism group is the trivial group. Its fundamental representation is the 248-dimensional adjoint representation.

Real forms

As well as the complex Lie group E₈, of complex dimension 248 or real dimension 496, there are 3 real forms of the group, all of real dimension 248. There is one compact one (which is usually the one meant if no other information is given), one split one, and a third one.

Constructions

One can construct the (compact form of the) $E_{8}$ group as the automorphism group of the corresponding $E_{8}$ Lie algebra. This algebra has a 120-dimensional subalgebra $\operatorname {so} (16)$ generated by $J_{ij}$ as well as 128 new generators $Q_{a}$ that transform as a Weyl-Majorana spinor of $\operatorname {spin} (16)$ . These statements determine the commutators

[J_{ij},J_{k\ell }]=\delta _{jk}J_{i\ell }-\delta _{j\ell }J_{ik}-\delta _{ik}J_{j\ell }+\delta _{i\ell }J_{jk}

as well as

[J_{ij},Q_{a}]={\frac {1}{4}}(\gamma _{i}\gamma _{j}-\gamma _{j}\gamma _{i})_{ab}Q_{b},

while the remaining commutator (not anticommutator!) is defined as

[Q_{a},Q_{b}]=\gamma _{ac}^{[i}\gamma _{cb}^{j]}J_{ij}.

It is then possible to check that the Jacobi identity is satisfied.

Geometry

The compact real form of E₈ is the isometry group of a 128-dimensional Riemannian manifold known informally as the 'octooctonionic projective plane' because it can be built using an algebra that is the tensor product of the octonions with themselves. This can be seen systematically using a construction known as the 'magic square', due to Hans Freudenthal and Jacques Tits.

In physics

The group E₈ frequently appears in string theory and supergravity, for example as the U-duality group of supergravity on an eight-torus (in its split form), or as a part of the gauge group of the heterotic string (the compact version).

Algebra

Dynkin diagram

Root system

All ${\begin{pmatrix}8\\2\end{pmatrix}}$ permutations of

(\pm 1,\pm 1,0,0,0,0,0,0).

and all of the following vectors

\left(\pm {1 \over 2},\pm {1 \over 2},\pm {1 \over 2},\pm {1 \over 2},\pm {1 \over 2},\pm {1 \over 2},\pm {1 \over 2},\pm {1 \over 2}\right)

for which the sum of all the eight coordinates is even.

There are 240 roots in all.

Simple roots:

(0,0,0,0,0,0,1,−1)

(0,0,0,0,0,0,1,1)

(0,0,0,0,0,1,−1,0)

(0,0,0,0,1,−1,0,0)

(0,0,0,1,−1,0,0,0)

(0,0,1,−1,0,0,0,0)

(0,1,−1,0,0,0,0,0)

(1/2,−1/2,−1/2,−1/2,−1/2,−1/2,−1/2,1/2)

Cartan matrix

{\begin{pmatrix}2&-1&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0\\0&-1&2&-1&0&0&0&-1\\0&0&-1&2&-1&0&0&0\\0&0&0&-1&2&-1&0&0\\0&0&0&0&-1&2&-1&0\\0&0&0&0&0&-1&2&0\\0&0&-1&0&0&0&0&2\end{pmatrix}}

Important maximal subgroups

Both $(E_{7}\times SU(2))/(Z/2Z)$ and $(E_{6}\times SU(3))/(Z/3Z)$ are maximal subgroups of $E_{8}$ .

The 248 adjoint representation of $E_{8}$ transforms under $SU(2)\times E_{7}$ as:

$(3,1)+(1,133)+(2,56)$

Since the adjoint representation can be described by the roots together with the generators in the Cartan subalgebra, we may see that decomposition by looking at these. In this description:

The $(3,1)$ consists of the roots (0,0,0,0,0,0,1,-1),(0,0,0,0,0,0,-1,1) and the Cartan generator corresponding to the last dimension.

The $(1,133)$ consists of all roots with (1,1),(-1,-1),(0,0),(-1/2,-1/2) or (1/2,1/2) in the last two dimensions, together with the Cartan generators corresponding to the first 7 dimensions.

The $(2,56)$ consists of all roots with permutations of (1,0),(-1,0) or (1/2,-1/2) in the last two dimensions.

The 248 adjoint representation of $E_{8}$ transforms under $SU(3)\times E_{6}$ as:

$(8,1)+(1,78)+(3,27)+({\bar {3}},{\bar {27}})$

We may again see the decomposition by looking at the roots together with the generators in the Cartan subalgebra. In this description:

The $(8,1)$ consists of the roots with permutations of (1,-1,0)in the last three dimensions, together the Cartan generator corresponding to the last two dimensions.

The $(1,78)$ consists of all roots with (0,0,0),(-1/2,-1/2,-1/2) or (1/2,1/2,1/2) in the last three dimensions, together with the Cartan generators corresponding to the first 6 dimensions.

The $(3,27)$ consists of all roots with permutations of (1,0,0),(1,1,0) or (-1/2,1/2,1/2) in the last three dimensions.

The $({\bar {3}},{\bar {27}})$ consists of all roots with permutations of (-1,0,0),(-1,-1,0) or (1/2,-1/2,-1/2) in the last three dimensions.

Importance in physics

In string theory, the gauge group of one of the two supersymmetric versions of the heterotic string is $E_{8}\times E_{8}$ .

One way to incorporate the standard model in the heterotic string includes the breaking of E₈ to its maximal subgroup $(SU(3)\times E_{6})/(Z/3Z)$ .

References

John Baez, The Octonions, Section 4.6: E₈, Bull. Amer. Math. Soc. 39 (2002), 145-205. Online HTML version at

http://math.ucr.edu/home/baez/octonions/node19.html.