Tag 0006
\(\langle \frac{1}{2} \rangle\)
<1/2>·A1*
Invariants
| Rank | 1 |
|---|---|
| Signature \((n_+, n_-)\) | (1, 0) |
| Determinant | \(\tfrac{1}{2}\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}\).
| \(e_{1}\) | |
|---|---|
| \(e_{1}\) | 1/2 |
[[1/2]]
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(\tfrac{1}{2}\) |
|---|---|
| Kissing number | 2 |
| Order of the isometry group | \(2 = 2\) |
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | yes |
|---|---|
| \(L = \mathbb{Z}\Phi_S(L)\) for | \(S = \{\tfrac{1}{2}\}\) |
| Number of roots | 2 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong \langle \tfrac{1}{2} \rangle\) |
| Rank of \(R(L)\) | 1 |
| \([L : R(L)]\) | 1 |
| \(R(L)\) is primitive in \(L\) | yes |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(A_{1}\) | \(\{\tfrac{1}{2}\}\) | 2 | \(\cong \langle \tfrac{1}{2} \rangle\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{1}\) | \(\tfrac{1}{2}\) |