Named lattices
Notation for the named lattices of the catalogue, and the chosen basis of each one.
Each name is built from the lattices below by orthogonal sums and twists \(L(k)\).
Diagonal lattices
\(\langle a \rangle\) is the lattice of rank 1 with basis \(e_1\) and \(b(e_1, e_1) = a\). \(\mathrm{I}_{p,q}\) is the orthogonal sum of \(p\) copies of \(\langle 1 \rangle\) and \(q\) copies of \(\langle -1 \rangle\). \(\mathrm{I}_{n,0}\) is also written \(\mathbb{Z}^n\): its basis \(e_1, \dots, e_n\) has \(b(e_i, e_j) = \delta_{ij}\).
The hyperbolic plane
\(U\) is the lattice with basis \(e, f\) and \(b(e, e) = b(f, f) = 0\), \(b(e, f) = 1\).
Root lattices
\(X_n\), for \(X_n\) one of \(A_n\) (\(n \geq
1\)), \(D_n\) (\(n \geq 4\)), \(E_6\), \(E_7\), \(E_8\), is the root lattice of that type: \(e_1, \dots, e_n\) is a basis of simple
roots, numbered as SageMath’s CartanMatrix numbers them,
and \(b(e_i, e_j)\) is the entry \((i, j)\) of the Cartan matrix.
\(\widetilde{X}_n\) is the lattice
of rank \(n + 1\) of the affine root
system of that type: its basis \(e_1, \dots,
e_{n+1}\) is a basis of simple roots, numbered as SageMath’s
CartanMatrix numbers them, and \(b(e_i, e_j)\) is the entry \((i, j)\) of the affine Cartan matrix.
For \(n\) even, \(D_n^+ := D_n \cup (h + D_n) \subset \mathbb{Q}^n\), where \(D_n = \{x \in \mathbb{Z}^n : x_1 + \dots + x_n \text{ even}\}\), \(h = (\tfrac{1}{2}, \dots, \tfrac{1}{2})\), and the form is \(b(x, y) = \sum_i x_i y_i\).
Dual lattices
\(L^*\) is the dual lattice of \(L\). For \(L\) in the catalogue, the chosen basis of \(L^*\) is \(e_1^*, \dots, e_n^*\) with \(b(e_i^*, e_j) = \delta_{ij}\), dual to the chosen basis of \(L\).