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Lecture 15: Dynkin Diagrams and Subgroups of Lie Groups

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Lecture 15: Dynkin Diagrams and Subgroups of Lie Groups Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage Lecture 15: Dynkin Diagrams and subgroups of Lie groups Daniel Bump May 26, 2020 Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage The rank two root systems The rank two root systems are, in the Cartan classification A2, C2, G2 and A1 × A1. Lie groups representing these are SL(3), Sp(4), G2 (the automorphism group of the octonions and SL(2) × SL(2). We will study general root systems by finding rank two root systems inside them, so let us take a closer look at the rank two root systems. We will denote by fα1; ··· ; αrg the simple roots. (In this section r = 2.) We will also introduce α0, the negative of the highest root which we may call the affine root. We proved if αi, αj are simple roots then hαi; αji 6 0. This remains true if we include α0 Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage The A2 root system α2 α1 α0 The shaded area is the positive Weyl chamber C+. The weight lattice is indicated as lighter dots. The root lattice has index 3 in the SU(3) weight lattice. Positive roots are red. If all roots have the same length, the root system is called simply-laced. The A2 root system is simply-laced. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage The C2 root system α2 α0 α1 This is the Sp(4) root system. The simple roots are α1 = (1; −1) and α2 = (0; 2). The root lattice has index two in the Sp(4) weight lattice, which we are identifying with Z2. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage The B2 root system α2 α0 α1 The SO(5) or spin(5) root system is accidentally isomorphic to the Sp(4) root system. The SO(5) weight lattice is Z2. The spin(5) weight lattice is 2 2 1 1 Z ⊕ Z + ( 2 ; 2 ) Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage The G2 root system α2 α1 α0 This time the root lattice equals the weight lattice. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage The A1 × A1 root lattice α1 α1 This is the reducible root system for SU(2) × SU(2). There is no affine root. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage The Dynkin diagram The Dynkin diagram is a graph whose vertices are the simple roots. Draw an edge connecting αi to αj if they are not orthogonal. For the extended Dynkin diagram, we add a node for α0. We often use a dashed line for connections of α0. Here is the extended Dynkin diagram for A3: α0 α1 = (1; −1; 0; 0) α2 = (0; 1; −1; 0) α3 = (0; 0; 1; −1) = (−1 0 0 1) α1 α2 α3 α0 ; ; ; Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage Double and triple bonds If αi and αj have different lengths, we connect them by: p a double bond if their root lengths are in the ratio 2; p a triple bond if their root lengths are in the ratio 3. The triple bond only occurs with G2. Here are the angles of the roots: bond angle example π no bond 2 SU(2) × SU(2) 2π single bond 3 SU(3) 3π double bond 4 Sp(4) 5π triple bond 6 G2 Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage The direction of the arrow If the roots are connected by a double or triple bond, they have different lengths. We draw an arrow from the long root to the short root. Here are the extended Dynkin diagram of type Bn and Cn: α0 α1 α2 α3 αn 2 αn 1 αn − − α0 α1 α2 α3 αn 2 αn 1 αn − − Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage What we learn from Dynkin diagrams The Dynkin diagram shows the relations between the simple roots. The extended Dynkin diagrams adds the affine root. From the Dynkin diagram we may read off: Generators and relations for the Weyl group; All Levi subgroups; From the extended Dynkin diagram we may read off: Generators and relations for the affine Weyl group; More general Lie subgroups Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage Type Dn α0 αn 1 − α1 α2 α3 αn 3 αn 2 − − αn The group D4 = spin(8) is particularly interesting. Here is its extended Dynkin diagram: α0 α1 α3 α2 α4 Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage Triality Another use of the Dynkin diagram is to make manifest the outer automorphisms of a Lie group. Symmetries of the Dynkin diagram may be realized as automorphisms of the group in its simply-connected form. The D4 Dynkin diagram has an automorphism of degree 3. α4 α2 α1 α3 This is an automorphism of the simply-connected group spin(8) or the adjoint form PGSO(8). Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage Triality (continued) The group spin(2k) has two irreducible representations of degree 2k−1 called the spin representations. It also has an irreducible representation of degree 2k, the standard representation. If k = 4, then 2k = 2k−1 = 8. Thus spin(8) has three irreducible representations of degree 8. These are permuted by triality. The reason is that the center of spin(8) is Z2 × Z2. Triality acts on the center and the kernel Z2 of the homomorphism spin(8) ! SO(8) is not invariant under triality. The fixed subgroup of this automorphism is the exceptional group G2, the automorphism of the octonions. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage Exceptional groups We will at least give the extended Dynkin diagrams for the exceptional types G2, F4, E6, E7 and E8. Here is G2: α0 α1 α2 There are two conventions for the ordering of the roots, due to Dynkin and Bourbaki. They differ in the exceptional groups. We are following Bourbaki. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage The exceptional group F4 α0 α1 α2 α3 α3 The group F4 is the next exceptional group. It is the automorphism group of a 27-dimensional (nonassociative) Jordan algebra discovered by A. A. Albert that is also closely related to the exceptional groups E6; E7 and E8. The exceptional group G2 is a subgroup. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage The exceptional groups E6, E7 and E8 α0 α2 α1 α3 α4 α5 α6 α2 α0 α1 α3 α4 α5 α6 α7 α2 α0 α1 α3 α4 α5 α6 α7 α8 Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage Levi subgroups One application of the Dynkin diagram and extended Dynkin diagram is to envision embeddings of Lie groups. Many maximal subgroups can be visualized instantly. The easiest case is that of a Levi subgroup. Let us choose a subset S of the simple roots and consider the complex Lie algebra generated by X±α; α 2 S: This is a Levi subgroup of the complex Lie group GC. (If we want we can intersect it with the compact Lie group G.) Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage Levi decomposition of parabolics Levi subgroups appear as Levi decompositions of parabolic subgroups. A subgroup P containing the Borel subgroup B (of GC) whose Lie algebra is tC ⊕ Xα + αM2Φ is called a parabolic subgroup. It is a semidirect product of a normal unipotent group and a parabolic subgroup. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage For example, let G = GL(4), S = fα1; α3g. Let ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ P = 8 9 : 0 ∗ ∗1 > > <B ∗ ∗C= B C >@ A> This has a decomposition:>P = MU with U;>normal: ∗ ∗ 1 ∗ ∗ ∗ ∗ 1 ∗ ∗ M = 8 9 ; U = 8 9 : 0 ∗ ∗1 0 1 1 > > > > <B ∗ ∗C= <B 1C= B C B C >@ A> >@ A> The subgroup:> M is a Levi subgroup.;> The:> group U is called;> the unipotent radical of P. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage Levi subgroups from Dynkin diagrams Starting from the Dynkin diagram of G, if we erase one or more nodes, we obtain the Dynkin diagram of a Levi subgroup. In the above example, the Dynkin diagram of GL(4) is of Type A3. α1 α2 α3 After selecting S = fα1; α3g, that is, erasing the middle node, we obtain the Dynkin diagram of the Levi subgroup GL(2) × GL(2), of type A1 × A1: α1 α3 All Levi subgroups can be determined easily from the Dynkin diagram. A Levi subgroup may or may not be a maximal subgroup. Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage Subgroups from the extended Dynkin diagram If we erase one node of the extended Dynkin diagram, we typically obtain the Dynkin diagram of a subgroup that is often a maximal subgroup. Here is the extended Dynkin diagram of SO(9) (Type B4): α0 α1 α2 α3 α4 Erasing the root α4 gives the Dynkin diagram of type D4 and we have obtained the embedding SO(8) ! SO(9); Rank 2 Root Systems Dynkin Diagrams Levi Subgroups Extended DD Root Folding Sage Review: convex sets of roots and Lie subalgebras In Lecture 7 we considered a subset S of Φ [ f0g such that α; β 2 S; α + β 2 Φ [ f0g ) α + β 2 S: (∗) We will call such a set convex.
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