Roots

Let \(L\) be a lattice.

A root of \(L\) is a primitive \(r \in L\) with \(b(r, r) \neq 0\) and \(s_r \in O(L)\), where \(s_r(x) = x - \frac{2 b(x, r)}{b(r, r)} r\); equivalently, \(2 b(r, L) \subseteq b(r, r) \mathbb{Z}\). \(\Phi(L)\) is the set of roots of \(L\), and \(\Phi_S(L) := \{r \in \Phi(L) : b(r, r) \in S\}\) for \(S \subseteq \mathbb{Q}\).

\(L\) is an \(S\)-root lattice when \(L = \mathbb{Z}\Phi_S(L)\), and a root lattice when \(L = \mathbb{Z}\Phi(L)\).

For the twist \(L(k)\), \(\Phi(L(k)) = \Phi(L)\) and \(\Phi_{kS}(L(k)) = \Phi_S(L)\).

\(E_8(2)\) is a \(\{4\}\)-root lattice.

Let \(L\) be integral and nondegenerate, and \(r \in \Phi(L)\) with \(b(r, L) = d\mathbb{Z}\), \(d > 0\). Then \(b(r, r) \in \{\pm d, \pm 2d\}\), and \(d\) divides the exponent of the discriminant group. Each \(r \in L\) with \(b(r, r) \in \{\pm 1, \pm 2\}\) is a root.

Root sublattices

The root sublattice of \(L\) with norms in \(S\) is \(R_S(L) := \mathbb{Z}\Phi_S(L)\), the sublattice generated by \(\Phi_S(L)\), and \(R(L) := R_{\mathbb{Q}}(L) = \mathbb{Z}\Phi(L)\).

\(R_S(L)\) is stable under \(O(L)\). \(L\) is a root lattice exactly when \([L : R(L)] = 1\), and the index is finite exactly when \(R(L)\) has the rank of \(L\).

\(R(L)\) is a root lattice.

A root \(r\) of \(L\) is primitive in \(R(L)\), and \(s_r(x) \in x + \mathbb{Z}r\) for each \(x \in L\), so \(r \in \Phi(R(L))\).

Let \(Q_S(L)\) be the set of the sublattices \(M \subseteq L\) with \(M = \mathbb{Z}(M \cap \Phi_S(L))\), ordered by inclusion. \(R_S(L)\) is the greatest element of \(Q_S(L)\), and each \(M \in Q_S(L)\) is an \(S\)-root lattice.

Each \(M \in Q_S(L)\) is generated by elements of \(\Phi_S(L)\), so \(M \subseteq R_S(L)\). \(M \cap \Phi(L) \subseteq \Phi(M)\), by the argument of the proposition above.

For primitive \(v \in L\) with \(b(v, v) \neq 0\), \(\mathbb{Z}v\) is a root lattice, and \(v\) can fail to be a root of \(L\).

The page of a lattice states the index \([L : R(L)]\), and the quotient when \(R(L)\) is not primitive in \(L\).

\(R(\mathbb{Z}^n) = \mathbb{Z}^n\), and \(R_{\{2\}}(\mathbb{Z}^n) = D_n\) has index \(2\) in \(\mathbb{Z}^n\) for \(n \geq 2\). For \(U\), \(R(U) = \mathbb{Z}(e + f) \oplus \mathbb{Z}(e - f) \cong \langle 2 \rangle \oplus \langle -2 \rangle\) has index \(2\) in \(U\).

\(D_{12}^+\) is generated by \(D_{12}\) and \(h = (\tfrac{1}{2}, \dots, \tfrac{1}{2})\). \(D_{12}^+\) is unimodular, so each root \(r\) has \(b(r, r) \in \{1, 2\}\). Each vector with coordinates in \(\tfrac{1}{2} + \mathbb{Z}\) has \(b(x, x) \geq 3\), so \(\Phi(D_{12}^+)\) is the set of the 264 vectors \(\pm e_i \pm e_j\), \(i \neq j\), and \([D_{12}^+ : R(D_{12}^+)] = [D_{12}^+ : D_{12}] = 2\).

Let \(P_S(L) \subseteq Q_S(L)\) be the subset of the primitive sublattices. \(P_S(L)\) has a greatest element exactly when \(R_S(L)\) is primitive in \(L\). Otherwise its maximal elements can be non-isometric.

\(M_1 = \{x \in D_{12}^+ : x_1 = 0\}\) and \(M_2 = \{x \in D_{12}^+ : x_1 = x_2 + \dots + x_{12}\}\) are primitive of rank 11, the roots of \(D_{12}^+\) in each one generate it, and each one is maximal in \(P_{\mathbb{Q}}(D_{12}^+)\): the only primitive sublattice of rank 12 is \(D_{12}^+\), which is not in \(P_{\mathbb{Q}}(D_{12}^+)\). \(M_1 \cong D_{11}\) has determinant 4, and \(M_2 \cong A_{11}\) has determinant 12: a vector of \(D_{12}^+\) with coordinates in \(\tfrac{1}{2} + \mathbb{Z}\) has an even sum of coordinates, and \(x_1 = x_2 + \dots + x_{12}\) makes the sum \(2x_1\) odd, so \(M_2 \subseteq \mathbb{Z}^{12}\).

The root lattices contained in \(D_{12}^+\) have no greatest element. With \(A' = \{x \in D_{12} : \sum_i x_i = 0\} \cong A_{11}\), both \(D_{12}\) and \(\mathbb{Z}h \oplus A'\) are root lattices in \(D_{12}^+\). \(\mathbb{Z}h \oplus A'\) has rank 12 and determinant 36, and \(h\) is a root of it but not of \(D_{12}^+\).

Hyperbolic lattices

Let \(L\) have signature \((1, n)\) and \(S \subseteq \mathbb{Q}_{<0}\); for signature \((n, 1)\), take \(S \subseteq \mathbb{Q}_{>0}\). The group \(W_S(L)\) generated by the reflections \(s_r\), \(r \in \Phi_S(L)\), is a discrete reflection group of the hyperbolic space \(\mathbb{H}^n\), the image of \(\{x \in L \otimes_{\mathbb{Z}} \mathbb{R} : b(x, x) > 0\}\) in \(\mathbb{P}(L \otimes_{\mathbb{Z}} \mathbb{R})\). Let \(\mathcal{P}\) be a fundamental polyhedron of \(W_S(L)\) with walls \(\alpha_i^\perp\), \(\alpha_i \in \Phi_S(L)\).

\(R_S(L) = \sum_i \mathbb{Z}\alpha_i\), so \(L\) is an \(S\)-root lattice exactly when the \(\alpha_i\) generate \(L\).

Each mirror of \(W_S(L)\) is \(w(\alpha_i^\perp)\) for some \(w \in W_S(L)\) and some \(i\). For \(r \in \Phi_S(L)\) with \(r^\perp = w(\alpha_i^\perp)\), \(r\) and \(w\alpha_i\) are proportional and both are primitive, so \(r = \pm w\alpha_i\). \(W_S(L)\) is generated by the \(s_{\alpha_j}\), and \(s_{\alpha_j}(x) \in x + \mathbb{Z}\alpha_j\), so \(w\alpha_i \in \sum_j \mathbb{Z}\alpha_j\).

Definite lattices

A positive definite \(\{2\}\)-root lattice is an orthogonal sum of lattices \(A_n\), \(D_n\), \(E_6\), \(E_7\) and \(E_8\) (Conway and Sloane, Sphere Packings, Lattices and Groups, 3rd edition, Chapter 4, §3).

The database states the ADE type of \(\Phi_{\{2\}}(L)\) for each definite integral lattice, with the sign convention for negative definite \(L\).

Let \(L\) be definite. \(\Phi(L)\) is a reduced root system in \(R(L) \otimes_{\mathbb{Z}} \mathbb{R}\). It is the union of irreducible components, orthogonal in pairs, of types \(A_n\) (\(n \geq 1\)), \(B_n\) (\(n \geq 2\)), \(C_n\) (\(n \geq 3\)), \(D_n\) (\(n \geq 4\)), \(E_6\), \(E_7\), \(E_8\), \(F_4\) and \(G_2\).

\(\Phi(L)\) is finite, \(O(L)\) permutes it, \(2 b(r, r') / b(r, r) \in \mathbb{Z}\) for \(r, r' \in \Phi(L)\), and the multiples of a root \(r\) in \(\Phi(L)\) are \(\pm r\), because a root is primitive.

\(\Phi(\mathbb{Z}^n)\) is of type \(B_n\) for \(n \geq 2\), with the roots \(\pm e_i\) and \(\pm e_i \pm e_j\), and \(\Phi(A_2)\) is of type \(G_2\).

The page states each component \(\Phi_c\) of \(\Phi(L)\) with its type, its scale \(k\) and its simple roots in the chosen basis. Each short root \(r\) of \(\Phi_c\) has \(b(r, r) = 2k\). \(R(L)\) is the orthogonal sum of the sublattices \(\mathbb{Z}\Phi_c\), and the simple roots are a basis of it. \(\mathbb{Z}\Phi_c\) is isometric to \(X_n(k)\) for a type \(X_n\) among \(A_n\), \(D_n\), \(E_n\); to \(\mathbb{Z}^n(2k)\) for \(B_n\); to \(D_n(k)\) for \(C_n\), with \(D_3 \cong A_3\); to \(D_4(k)\) for \(F_4\); and to \(A_2(k)\) for \(G_2\).

Lattices that are not definite

\(\Phi(L)\) can be infinite. The page states roots \(r_1, r_2, \dots\) that generate \(R(L)\). When the \(r_i\) generate \(L\), \(L = R(L)\). When they do not, the notes prove that each root of \(L\) is in \(\sum_i \mathbb{Z}r_i\), and the page states lattices \(M_1, M_2, \dots\) of the catalogue and the matrix of an isometric embedding \(M_1 \oplus M_2 \oplus \cdots \to L\) with image \(R(L)\).

Labels and the page of a lattice

The label root lattice means that \(L = R(L)\). The label not a root lattice means that \(L \neq R(L)\). For a lattice with neither label, the question is not decided. The page of a lattice states a lattice of the catalogue isometric to \(R(L)\), its rank, the index \([L : R(L)]\) and whether \(R(L)\) is primitive in \(L\), with the quotient when it is not. Each of these rows is not decided when \(R(L)\) is not stated. For a root lattice the page states a set \(S\) with \(L = \mathbb{Z}\Phi_S(L)\). Several sets \(S\) can occur: \(\mathbb{Z}^n = \mathbb{Z}\Phi_S(\mathbb{Z}^n)\) for each \(S \ni 1\).