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Classical root systems in coordinates

Example

For $n \geq 2$ , in the standard coordinates $ε_{1}, \dots, ε_{n}$ of $R^{n}$ (and the sum-zero hyperplane for type $A$ ):

A_{n - 1} = {ε_{i} - ε_{j} : i \neq j}, B_{n} = {\pm ε_{i}} \cup {\pm ε_{i} \pm ε_{j} : i < j},

C_{n} = {\pm 2 ε_{i}} \cup {\pm ε_{i} \pm ε_{j} : i < j}, D_{n} = {\pm ε_{i} \pm ε_{j} : i < j} .

Each is a reduced crystallographic Euclidean root system with the standard simple roots and Dynkin diagram, and each is the root system computed from the corresponding classical matrix Lie algebra.

Facts & Assumptions

Given: An integer $n \geq 2$ , the standard orthonormal basis $ε_{1}, \dots, ε_{n}$ of $R^{n}$ , and the four displayed sets.

[L1]

A reduced crystallographic root system is a finite spanning set of nonzero vectors that is closed under its root reflections, has integral Cartan integers, and meets each root line in exactly the two signs (Reduced crystallographic Euclidean root system).

[L2]

The root systems of the classical matrix Lie algebras with diagonal Cartan subalgebras are these same sets, with one-dimensional root spaces (Root systems of the classical complex Lie algebras).

[L3]

In the cited coordinate models, the standard simple roots are $ε_{i} - ε_{i + 1}$ for $A_{n - 1}$ ; $ε_{1} - ε_{2}, \dots, ε_{n - 1} - ε_{n}, ε_{n}$ for $B_{n}$ ; $ε_{1} - ε_{2}, \dots, ε_{n - 1} - ε_{n}, 2 ε_{n}$ for $C_{n}$ ; and, for $D_{n}$ with $n \geq 3$ , $ε_{1} - ε_{2}, \dots, ε_{n - 2} - ε_{n - 1}, ε_{n - 1} - ε_{n}, ε_{n - 1} + ε_{n}$ . For $D_{2}$ the two simple roots are $ε_{1} - ε_{2}$ and $ε_{1} + ε_{2}$ .

Verification

technique · direct

1.1L1algebra

Each set is finite, omits $0$ , and is reduced. The differences $ε_{i} - ε_{n}$ span the sum-zero hyperplane for $A_{n - 1}$ ; $B_{n}$ and $C_{n}$ contain a nonzero multiple of every coordinate vector; and in $D_{n}$ , $(ε_{i} + ε_{j}) + (ε_{i} - ε_{j}) = 2 ε_{i}$ for any $j \neq i$ , which exists because $n \geq 2$ . Thus each set spans its stated Euclidean space.

1.2L1algebra

Reflection closure: $s_{ε_{i}}$ and $s_{2 ε_{i}}$ negate the $i$ th coordinate and preserve $B_{n}, C_{n}, D_{n}$ ; $s_{ε_{i} - ε_{j}}$ swaps coordinates $i, j$ and preserves all four sets; and $s_{ε_{i} + ε_{j}}$ swaps and negates those two coordinates and preserves $B_{n}, C_{n}, D_{n}$ . These are precisely the root reflections that occur in the displayed sets.

2.1L1step 1.1step 1.2algebra

Integrality: proportional pairs give Cartan integer $\pm 2$ . For nonproportional pairs, roots of squared length $2$ pair by $0$ or $\pm 1$ ; a short root of squared length $1$ in $B_{n}$ pairs by $0$ or $\pm 1$ ; and a long root $2 ε_{i}$ of squared length $4$ in $C_{n}$ pairs with a mixed root by $0$ or $\pm 2$ . Hence every Cartan integer is in ${0, \pm 1, \pm 2}$ . Together with steps 1.1 and 1.2, [L1] proves that all four displayed sets are reduced crystallographic root systems.

3.1L2L3step 2.1algebra∎

The listed simple roots of [L3] have the standard Cartan matrices of $A_{n - 1}, B_{n}, C_{n}, D_{n}$ (with $D_{2} = A_{1} ⊔ A_{1}$ ), giving the stated Dynkin diagrams; and [L2] identifies these coordinate sets as the root systems of ${s l}_{n}, {s o}_{2 n + 1}, {s p}_{2 n}, {s o}_{2 n}$ respectively.

Remarks

The following Dynkin schematics use the simple roots of [L3], numbered in their listed order. For $A_{n - 1}$ , $α_{i} = ε_{i} - ε_{i + 1}$ . For $B_{n}$ and $C_{n}$ the same formula holds for $i < n$ , with $α_{n} = ε_{n}$ and $α_{n} = 2 ε_{n}$ , respectively. For $D_{n}$ , $α_{i} = ε_{i} - ε_{i + 1}$ for $i < n$ and $α_{n} = ε_{n - 1} + ε_{n}$ . The schematics are drawn for $n \geq 5$ ; dots abbreviate intermediate chain vertices. An arrow on a double edge points toward the shorter simple root.

The smaller ranks are obtained by shortening the chains, with $A_{1}$ a single vertex. In $D_{3}$ the chain is $α_{2} - α_{1} - α_{3}$ ; in $D_{4}$ the fork vertex is $α_{2}$ . In $D_{2}$ , the roots $α_{1} = ε_{1} - ε_{2}$ and $α_{2} = ε_{1} + ε_{2}$ are orthogonal, so its two vertices are disconnected:

Indeed, the adjacent difference roots have squared length $2$ and inner product $- 1$ . The last $B_{n}$ pair has squared lengths $2, 1$ and inner product $- 1$ ; the last $C_{n}$ pair has squared lengths $2, 4$ and inner product $- 2$ . For $D_{n}$ , both fork edges have inner product $- 1$ , while the two terminal roots are orthogonal. Thus the double edges point in opposite directions in $B_{n}$ and $C_{n}$ , and the $D_{n}$ fork has only single edges, as in step 3.1. The verification above is independent of these illustrations.

Depends on

Used by

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Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Chapter II (standard reference, not scraped)
Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I, Lectures 19-24 (standard reference, not scraped)