Section 5.6 Examples: Rank Two
| \(\aa_2\) | \(\begin{aligned} \bullet\!-\!\bullet \end{aligned}\) |
| \(\bb_2\) | \(\begin{aligned} \bullet=\!\!\Leftarrow\bullet \end{aligned}\) |
| \(\cc_3\) | \(\begin{aligned} \bullet\Rightarrow\!\!=\bullet \end{aligned}\) |
| \(\gg_2\) | \(\begin{aligned} \bullet\Rrightarrow\!\!\equiv\bullet \end{aligned}\) |
The rank \(r\) of a simple Lie algebra is the dimension of its Cartan subalgebra. Since the roots are labeled by their eigenvalues under elements of this subalgebra, the root diagram is \(r\)-dimensional.
The first Lie algebra we encountered, in Section 1.1, was \(\so(2)\text{,}\) which is one-dimensional. This Lie algebra has no roots! Furthermore, the Killing form on \(\so(2)\) is degenerate. So \(\so(2)\) is not normally considered to be a simple Lie algebra.
The next Lie algebra we encountered, in Section 1.2–1.3, was \(\so(3)\cong\su(2)\text{,}\) which has just one simple root. As constructed in Section 4.2, the root diagram of \(\su(2)\) is one-dimensional, as shown in Figure 4.2.1. This Lie algebra is classified as \(\aa_1\text{.}\)
With two simple roots, the root diagrams are more interesting. As discussed in Section 5.5, the angles between simple roots can only be \(\frac\pi2\text{,}\) \(\frac{2\pi}3\text{,}\) \(\frac{3\pi}4\text{,}\) or \(\frac{5\pi}6\text{,}\) with corresponding restrictions on the relative magnitudes of the roots. It is now straightforward to list all of the possiblities; the corresponding Dynkin diagrams are shown in Table 5.6.1. The root diagrams are shown below, where in each case, the Cartan algebra is represented by a point at the origin, but is a two-dimensional vector space.
If the roots are orthogonal, then we obtain the first root diagram in Figure 5.6.2, showing two copies of the root diagram for \(\aa_1\cong\su(2)\) on top of each other. Since there is no nontrivial overlap between the two copies of \(\su(2)\text{,}\) this Lie algebra is not simple, but rather a direct sum. This Lie algebra is classified as \(\dd_2\text{,}\) and we have \(\dd_2\cong\su(2)\oplus\su(2)\cong\so(4)\text{.}\) For higher rank, it turns out that \(\dd_r\cong\so(2r)\text{,}\) which is simple (for \(r\ge2\)).
If the roots are at an angle of \(\frac{2\pi}3\text{,}\) then the root diagram is the hexagon shown in the second diagram in Figure 5.6.2, which we recognize from Section 4.3 as the root diagram of \(\su(3)\text{.}\) This Lie algebra is classified as \(\aa_2\text{,}\) and in general \(\aa_r\cong\su(r+1)\text{.}\)
If the roots are at an angle of \(\frac{3\pi}4\text{,}\) their magnitudes must have a ratio of \(\sqrt2\text{;}\) one root is “short”, and the other is “long”. Fixing the first root along the horizontal axes, both cases are shown in Figure 5.6.3. The first diagram corresponds to the classification \(\bb_2\text{,}\) and the second to \(\cc_2\text{,}\) which turn out to be associated with the odd orthogonal Lie algebras (\(\bb_r\cong\so(2r+1)\)) and symplectic Lie algebras (\(\cc_r\cong\sp(2r)\cong\su(r,\HH)\)), respectively. In this case, however, these two diagrams are clearly equivalent, since the overall scale is arbitrary, as is the orientation. Thus, we have the sequence of identifications
Finally, if the roots are at an angle of \(\frac{5\pi}6\text{,}\) their magnitudes must have a ratio of \(\sqrt3\text{.}\) This Lie algebra is classified as \(\gg_2\text{,}\) and its root diagram is shown in Figure 5.6.4.
These root diagrams can be used to examine the subalgebra structure of these Lie algebras. For instance, \(\su(3)\subset\gg_2\text{.}\) Do you see why?
Activity 5.6.1. Subalgebras of \(\gg_2\).
Determine which roots of \(\gg_2\) correspond to a \(\su(3)\) subalgebra.
Careful! There are two hexagons, one with short roots, the other with long roots. Only one of these hexagons represents a closed subalgebra. Which one does so?
Some short roots combine (through vector addition, corresponding to commutators in the Lie algebra) to form long roots, so the “short” hexagon does not form a closed algebra. However, long roots only combine to form other long roots, so the “long” hexagon does correspond to a Lie subalgebra, which is then clearly a copy of \(\su(3)\text{.}\)