2 Foundations: E8 Lattice
The \(E_8\) root system is the largest exceptional simple Lie algebra. We formalize its lattice structure in \(\mathbb {R}^8\).
2.1 Euclidean Space Setup
Let \(\mathbb {R}^8\) denote the 8-dimensional Euclidean space with standard inner product.
The standard basis vectors \(e_i\) for \(i \in \{ 0, \ldots , 7\} \) satisfy \(\langle e_i, e_j \rangle = \delta _{ij}\).
For all \(i, j \in \{ 0, \ldots , 7\} \):
For \(v \in \mathbb {R}^8\): \(\| v \| ^2 = \sum _{i=0}^{7} v_i^2\)
For \(v, w \in \mathbb {R}^8\): \(\langle v, w \rangle = \sum _{i=0}^{7} v_i w_i\)
2.2 E8 Lattice Definition
A vector \(v \in \mathbb {R}^8\) has all integer coordinates if \(v_i \in \mathbb {Z}\) for all \(i\).
A vector \(v \in \mathbb {R}^8\) has all half-integer coordinates if \(v_i \in \mathbb {Z}+ \frac{1}{2}\) for all \(i\).
A vector \(v\) has even sum if \(\sum _{i=0}^{7} v_i \in 2\mathbb {Z}\).
The \(E_8\) lattice consists of all \(v \in \mathbb {R}^8\) satisfying either:
All coordinates are integers with even sum, or
All coordinates are half-integers with even sum
2.3 Lattice Properties
For integers \(n_0, \ldots , n_7\): \(\left(\sum _i n_i^2\right) \mod 2 = \left(\sum _i n_i\right) \mod 2\)
Since \(n^2 \equiv n \pmod{2}\) (as \(n(n-1)\) is always even), the result follows by summing over all coordinates.
For \(v, w \in E_8\): \(\langle v, w \rangle \in \mathbb {Z}\)
Case analysis on integer/half-integer coordinates with parity arguments.
For \(v \in E_8\): \(\| v \| ^2 \in 2\mathbb {Z}\)
By Lemma 2.10, sum of squared integers has same parity as sum. For half-integers, \(\sum (n_i + 1/2)^2 = \sum n_i^2 + \sum n_i + 2\), which is even.
For \(v, w \in E_8\): \(v - w \in E_8\)
For a root \(\alpha \) with \(\langle \alpha , \alpha \rangle = 2\), the Weyl reflection is:
For \(\alpha , v \in E_8\) with \(\langle \alpha , \alpha \rangle = 2\): \(s_\alpha (v) \in E_8\)
Since \(\langle v, \alpha \rangle \in \mathbb {Z}\) by Theorem 2.11 and \(E_8\) is closed under integer scaling and subtraction.