John Baez

Kostant on E8

February 12, 2008

Here are my lecture notes from Bertram Kostant's talk on E8.

1. Chopping e8 into 31 pieces of dimension 8

In what follows we work with the complex form of E8 and its Lie algebra e8. Choose a root system Δ for e8. Each root φ in Δ gives a root vector eφ in e8, and e8 is then a direct sum of its Cartan subalgebra h and the spans of these vectors eφ. The Cartan subalgebra is 8-dimensional, and there are 240 roots, so the dimension of e8 is 248.

Choose simple roots α1, ... α8. There is a special element w in h such that

w ⋅ αi = 1

for i = 1,...,8. This gives a 1-parameter subgroup exp(sw) in E8, where s ∈ C. An element of E8 is called principal if it is conjugate to one of the form exp(sw). There is a unique element of order 2 of the form exp(sw), so we get a distinguished conjugacy class of order-2 elements in E8, namely the principal ones. There's a copy of the group (Z/2)5 sitting inside this conjugacy class. The group (Z/2)5 has 32 characters, that is, homomorphisms

χ : (Z/2)5 → U(1).

31 of these characters are nontrivial. Note that

248 = 31 × 8.

The nontrivial characters χj (j = 1,...,31) have an interesting property. Let

hj = {x ∈ e8 : Ad(a) x = χj(a) x for all a ∈ (Z/2)5 }

Then hj is a Cartan subalgebra of e8! And,

e8 = h1 ⊕ ⋅ ⋅ ⋅ ⊕ h31

So, the 248-dimensional Lie algebra e8 is a direct sum of 31 8-dimensional Cartan algebras.

2. How SL(2,32) acts to permute the 31 pieces of e8

E8 has a finite subgroup called the Dempwolf group, FDemp, with the above (Z/2)5 as a normal subgroup. In fact we have an exact sequence:

1 → (Z/2)5 → FDemp → SL(5,Z/2) → 1

Starting from the above (Z/2)5 subgroup of E8, we get a chain of subgroups

Z/2 ⊂ (Z/2)2 ⊂ (Z/2)3 ⊂ (Z/2)4 ⊂ (Z/2)5

The centralizer of Z/2 in E8 is D8 = Spin(16).

The centralizer of (Z/2)2 is D4 × D4, or more precisely, Spin(8) × Spin(8) mod the diagonal copy of Z/2.

We have a vector space decomposition

e8 = (so(8) ⊕ so(8))   ⊕   V8⊗V8   ⊕   S8+⊗S8+   ⊕   S8-⊗S8-

where V8, S8+ and S8- are the 8-dimensional "vector", "right-handed spinor" and "left-handed spinor" representations of Spin(8), respectively. These three representations are related by triality.

The elements of the Dempwolf group permute the 64-dimensional subspaces V8⊗V8, S8+⊗S8+ and S8-⊗S8- of e8.

Griess and Ryba have been studying finite subgroups of E8, for example:

Serre has too: The 5-dimensional vector space (Z/2)5 can be made into a field, F32, and invertible 2 × 2 matrices with determinant 1 having entries in this field form the group SL(2,32). This whole group sits inside E8. It contains the aforementioned copy of (Z/2)5 as matrices of the form

1   x
0   1

Proving this was hard!

The group SL(2,32) has cardinality

3 × 11 × 31 × 32

corresponding to something analogous to an Iwasawa decomposition G = KAN: the subgroup K has cardinality 3 × 11, the subgroup A has cardinality 31, and the subgroup N (consisting of the upper triangular matrices shown above) has cardinality 32.

The solvable subgroup of SL(2,32) has one irreducible 31-dimensional representation. The adjoint action of SL(2,32) on e8 acts to permute its 31 Cartan subalgebras according to this representation!

3. A product of two copies of the Standard Model gauge group in E8

The gauge group of the Standard model is usually said to be SU(3)×SU(2)×U(1), but this group has a Z/6 subgroup that acts trivially on all known particles. The quotient (SU(3)× SU(2)×U(1))/(Z/6) is isomorphic to S(U(3)×U(2)) - that is, the subgroup of SU(5) consisting of block diagonal matrices with a 3×3 block and a 2×2 block. So, this could be called the "true" gauge group of the Standard Model. Each generation of fermions transforms according to the obvious representation of S(U(3) × U(2)) on the exterior algebra ΛC5. This idea is the basis of the SU(5) grand unified theory. (For details, see week253.)

There is an element of order 11 in SL(2,32) ⊂ E8. The centralizer of this element in E8 is a product of two copies of the Standard Model gauge group:

S(U(3)×U(2)) × S(U(3)×U(2))

Let gi be the subspace of e8 consisting of linear combinations of root vectors eφ where the root φ is a sum of i simple roots. Let g-i be the analogous thing defined using the negatives of the simple roots. Then, the Lie algebra of the S(U(3)×U(2)) × S(U(3)×U(2)) subgroup of E8 is

g-22 ⊕ g-11 ⊕ g11 ⊕ g22

(There is also an element of order 3 in SL(2,32), whose centralizer in E8 is SU(9).)

4. A product of two copies of su(5) in e8

There's also a copy of (Z/5)3 in E8. If we think of this as a 3-dimensional vector space containing "lines" (1-dimensional subspaces), then it contains

1 + 5 + 52 = 31

lines. The centralizer in E8 of any such line is the group

(SU(5) × SU(5))/(Z/5)

This group is 48-dimensional. It has a 248-dimensional representation coming from the adjoint action on e8. This is the direct sum of the 48-dimensional subrepresentation coming from su(5) ⊕ su(5) ⊂ e8 and a representation of dimension

248 - 48 = 200

This 200-dimensional representation is

5 ⊗ 10   ⊕   5* ⊗ 10*   ⊕   10 ⊗ 5*   ⊕   10* ⊗ 5

where 5 is the defining representation of SU(5), 10 is its second exterior power, and 5* and 10* are the duals of these.

Is any of this useful in particle physics? In particular, what could the "doubling" mean?

(I thank Allen Knutson, Jacques Distler and Jan Saxl for correcting some errors in my lecture notes. There could be more, and they're all my fault.)

Text © 2008 John Baez
Image © 2007 John Stembridge