This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. The Korean original is the source of truth.
In §Root Systems we defined the root system \(\Phi\) of a semisimple Lie algebra \(\mathfrak{g}\) and verified that its symmetry is captured by the Weyl group. Also, in §Torus Action we examined the relationship between the maximal torus \(T\) of a compact Lie group \(G\) and the Weyl group \(W=N(T)/T\). In this post we classify Lie algebras via the structure of root systems and introduce the geometric object that arises naturally from this: the flag variety.
Dynkin Diagram
The structure of a root system \(\Phi\) is completely determined by the relations among simple roots. The Cartan matrix defined in §Root Systems, ⁋Definition 16 expresses this relation in matrix form, but we can grasp the structure of a root system more intuitively through visualization.
Definition 1 For a root system \(\Phi\) and a set of simple roots \(\Delta=\{\alpha_1,\ldots,\alpha_l\}\), the Dynkin diagram of \(\Phi\) is the graph defined as follows.
- Place one vertex for each simple root \(\alpha_i\).
- Place \(\lvert\langle\alpha_i,\alpha_j\rangle\rvert\) edges between the two vertices \(\alpha_i\) and \(\alpha_j\) (\(i\neq j\)).
- If \(\lvert\alpha_i\rvert\neq\lvert\alpha_j\rvert\), add an arrow pointing toward the longer root on the edge.
Since \(a_{ij}=\langle\alpha_i,\alpha_j\rangle\) in the Cartan matrix \(A=(a_{ij})\), one can think of the Dynkin diagram as encoding the information of the Cartan matrix in a graph. As we saw in §Root Systems, \(a_{ij}\leq 0\) and \(a_{ij}=0\) is equivalent to \(a_{ji}=0\), so the number of edges is determined symmetrically. Moreover, since \(a_{ij}\in\{0,-1,-2,-3\}\), there are at most three edges between any two vertices.
Example 2 Consider \(\Phi(A_n)\) from §Root Systems, ⁋Example 13. We may choose simple roots \(\alpha_i=e_i-e_{i+1}\) (\(1\leq i\leq n\)). Computing the inner products among them gives
\[(\alpha_i,\alpha_j)=\begin{cases}2 & i=j\\ -1 & \lvert i-j\rvert=1\\ 0 & \text{otherwise}\end{cases}\]Thus \(\langle\alpha_i,\alpha_j\rangle\) is \(2\) when \(i=j\), \(-1\) when \(\lvert i-j\rvert=1\), and \(0\) otherwise. Therefore the Dynkin diagram consists of \(n\) vertices connected in a chain, and since all roots have the same length there are no arrows.
The key properties of Dynkin diagrams are as follows.
Proposition 3 A Dynkin diagram is either connected or a disjoint union of connected components. Each connected component corresponds to an irreducible root system.
Proof
The connected components of a Dynkin diagram define a partition \(\Delta=\Delta_1\sqcup\cdots\sqcup\Delta_k\) of the simple roots. Considering the root subsystem \(\Phi_i\) generated by each \(\Delta_i\), by §Root Systems, ⁋Proposition 6 roots belonging to different \(\Delta_i\) are orthogonal. Thus \(\Phi=\Phi_1\sqcup\cdots\sqcup\Phi_k\), and each \(\Phi_i\) is irreducible.
Conversely, if the Dynkin diagram of an irreducible root system were not connected, the above argument would make it reducible, a contradiction.
Proposition 4 A Dynkin diagram contains no cycles. In other words, a Dynkin diagram is always a tree or a forest.
Proof
Suppose for contradiction that a cycle \(\alpha_{i_1},\ldots,\alpha_{i_k}=\alpha_{i_1}\) exists. Let \(k\) be the number of edges in the cycle. For each edge we have \(\langle\alpha_{i_j},\alpha_{i_{j+1}}\rangle\neq 0\), and therefore \(\langle\alpha_{i_j},\alpha_{i_{j+1}}\rangle\leq -1\).
Now consider \(\alpha=\sum_{j=1}^{k-1}\alpha_{i_j}\). Then
\[(\alpha,\alpha)=\sum_{j=1}^{k-1}(\alpha_{i_j},\alpha_{i_j})+2\sum_{j<\ell}(\alpha_{i_j},\alpha_{i_\ell})\]Since this is a cycle, each \(\alpha_{i_j}\) is connected to exactly two neighbors, and thus
\[(\alpha,\alpha)\leq 2(k-1)-2(k-1)=0\]This contradicts the positive-definiteness of the inner product.
Proposition 5 In a Dynkin diagram the total number of edges emanating from a single vertex does not exceed \(4\). That is, for any simple root \(\alpha\),
\[\sum_{\beta\in\Delta,\beta\neq\alpha}\lvert\langle\alpha,\beta\rangle\rvert\leq 4\]Proof
Let \(H_\alpha\) be the hyperplane orthogonal to the simple root \(\alpha\). Considering the simple roots \(\beta_1,\ldots,\beta_m\) adjacent to \(\alpha\), each \(\beta_i\) makes a distinct angle with \(H_\alpha\).
One can show that \(m\leq 3\) from the linear independence of the vectors obtained by projecting the \(\beta_i\) onto \(H_\alpha\). Moreover, for each \(\beta_i\) we have \(\lvert\langle\alpha,\beta_i\rangle\rvert\leq 3\), so the total sum does not exceed \(4\).
ADE Classification
Now let us examine the classification of irreducible root systems. The preceding propositions severely restrict the possible shapes of Dynkin diagrams.
Theorem 6 The Dynkin diagram of an irreducible root system is one of the following types.
- Classical types:
- \(A_n\) (\(n\geq 1\)): \(n\) vertices connected in a chain
- \(B_n\) (\(n\geq 2\)): add a double edge and arrow at one end of \(A_n\)
- \(C_n\) (\(n\geq 2\)): add a double edge and arrow in the opposite direction at one end of \(A_n\)
- \(D_n\) (\(n\geq 4\)): branch at one end of \(A_{n-1}\)
- Exceptional types:
- \(E_6, E_7, E_8\): special forms with 6, 7, and 8 vertices respectively
- \(F_4\): 4 vertices with a double edge in the middle
- \(G_2\): 2 vertices joined by a triple edge
The proof of this classification proceeds by systematically analyzing the conditions that a Dynkin diagram must satisfy. The key ideas are as follows.
- Because there are no cycles, it must be a tree.
- Because branching is restricted, the possible forms are limited.
- From the restriction on the number of edges, the positions of double and triple edges are determined.
We omit the detailed proof; what is important for understanding the theorem is knowing what features each type possesses.
Example 7 The classical Lie algebras corresponding to each type are as follows.
| Type | Lie algebra | Dimension |
|---|---|---|
| \(A_n\) | \(\mathfrak{sl}(n+1,\mathbb{C})\) | \(n(n+2)\) |
| \(B_n\) | \(\mathfrak{so}(2n+1,\mathbb{C})\) | \(n(2n+1)\) |
| \(C_n\) | \(\mathfrak{sp}(2n,\mathbb{C})\) | \(n(2n+1)\) |
| \(D_n\) | \(\mathfrak{so}(2n,\mathbb{C})\) | \(n(2n-1)\) |
The Lie algebras corresponding to the exceptional types \(E_6, E_7, E_8, F_4, G_2\) cannot be realized as classical matrix algebras. Their dimensions are \(78, 133, 248, 52, 14\) respectively.
Simply-Laced Root System
A root system in which all roots have the same length is called simply-laced. They enjoy special properties.
Definition 8 A root system \(\Phi\) is simply-laced if \(\lvert\alpha\rvert=\lvert\beta\rvert\) for all \(\alpha,\beta\in\Phi\). Equivalently, its Dynkin diagram has no double or triple edges.
The simply-laced root systems are exactly those of type \(A_n\), \(D_n\), \(E_6\), \(E_7\), \(E_8\); these are collectively called ADE type. ADE types appear in a variety of mathematical situations. For example, ADE patterns arise in the classification of du Val singularities, the symmetry groups of Platonic solids, and two-dimensional conformal field theory, among others.
Borel Subalgebra
Now let us look at subalgebras of a Lie algebra that are defined naturally from a root system. In §Root Systems, ⁋Definition 15 we defined the collection \(\Phi^+\) of positive roots. This amounts to choosing a Weyl chamber, and from this we can define a special subalgebra of the Lie algebra.
Definition 9 For a semisimple Lie algebra \(\mathfrak{g}\), a Cartan subalgebra \(\mathfrak{h}\), and a set of positive roots \(\Phi^+\), the Borel subalgebra is the following subalgebra.
\[\mathfrak{b}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Phi^+}\mathfrak{g}_\alpha\]Here \(\mathfrak{n}=\bigoplus_{\alpha>0}\mathfrak{g}_\alpha\) is called the nilradical of \(\mathfrak{b}\).
A Borel subalgebra contains all root spaces corresponding to positive roots and can be thought of as the largest “upper triangular” subalgebra containing the Cartan subalgebra.
Proposition 10 For a Borel subalgebra \(\mathfrak{b}\), the following hold.
- \(\mathfrak{b}\) is solvable.
- \(\mathfrak{b}\) is a maximal solvable subalgebra of \(\mathfrak{g}\).
- Every conjugate of \(\mathfrak{b}\) is a Borel subalgebra. That is, for any \(g\in G\), \(\Ad(g)\mathfrak{b}\) is the Borel subalgebra for some positive system \(\Phi'^+\).
Proof
(1) Consider the derived series of \(\mathfrak{b}\). We have \(\mathfrak{b}^{(1)}=[\mathfrak{b},\mathfrak{b}]=\mathfrak{n}\), and since \(\mathfrak{n}\) is nilpotent, \(\mathfrak{b}\) is solvable. Specifically, \(\mathfrak{n}\) has a structure analogous to the algebra of strictly upper triangular matrices.
(2) Suppose there is a solvable subalgebra \(\mathfrak{s}\) containing \(\mathfrak{b}\). By the root decomposition it must be of the form \(\mathfrak{s}=\mathfrak{h}\oplus\bigoplus_{\alpha\in S}\mathfrak{g}_\alpha\). If \(S\) does not contain some positive root then \(\mathfrak{s}\subset\mathfrak{b}\); if \(S\) contains a negative root then \(\mathfrak{s}\) is no longer solvable. Therefore \(\mathfrak{s}=\mathfrak{b}\).
(3) Since \(\Ad(g)\mathfrak{b}\) is again a maximal solvable subalgebra, by (2) above it is the Borel subalgebra for some positive system.
Borel Subgroup and Flag Variety
Now let us move to the Lie group perspective. For a complex semisimple Lie group \(G_\mathbb{C}\), we can consider the Lie subgroup corresponding to a Borel subalgebra \(\mathfrak{b}\).
Definition 11 The Borel subgroup \(B\) of a complex semisimple Lie group \(G_\mathbb{C}\) is the connected Lie subgroup corresponding to a Borel subalgebra \(\mathfrak{b}\).
\[\mathfrak{b}=\Lie(B)\]A Borel subgroup \(B\) is a maximal connected solvable subgroup of \(G_\mathbb{C}\). Now let us define the quotient space.
Definition 12 For a complex semisimple Lie group \(G_\mathbb{C}\) and its Borel subgroup \(B\), the flag variety is the following homogeneous space.
\[\mathcal{F}=G_\mathbb{C}/B\]The name flag variety comes from the fact that for \(\GL(n,\mathbb{C})\), \(\mathcal{F}\) coincides with the space of complete flags in \(\mathbb{C}^n\). In general the flag variety is a projective variety, and it has a deep relationship with the representation theory of \(G_\mathbb{C}\).
Example 13 Consider the case \(G_\mathbb{C}=\GL(n,\mathbb{C})\). The Borel subgroup \(B\) is the set of upper triangular matrices, and the flag variety \(\GL(n,\mathbb{C})/B\) is in bijection with the space of complete flags in \(\mathbb{C}^n\)
\[0=V_0\subset V_1\subset V_2\subset\cdots\subset V_n=\mathbb{C}^n,\qquad \dim V_i=i\]Specifically, \(gB\in\GL(n,\mathbb{C})/B\) corresponds to the flag \(V_i=\span\{ge_1,\ldots,ge_i\}\). This space is realized as a projective variety via the embedding
\[\GL(n,\mathbb{C})/B\hookrightarrow\mathbb{P}(\wedge^1\mathbb{C}^n)\times\mathbb{P}(\wedge^2\mathbb{C}^n)\times\cdots\times\mathbb{P}(\wedge^{n-1}\mathbb{C}^n)\]Connection with the Compact Form
Now let us examine the relationship between a compact Lie group \(G\) and its complexification \(G_\mathbb{C}\). This connection provides a bridge between the two perspectives — \(G/T\) for the compact group and \(G_\mathbb{C}/B\) for the complex group.
Definition 14 A compact form of a complex Lie group \(G_\mathbb{C}\) is a compact Lie group \(G\) satisfying the following conditions.
- \(G\) is a Lie subgroup of \(G_\mathbb{C}\).
- The Lie algebra \(\mathfrak{g}_0\) of \(G\) is a real form of \(\mathfrak{g}\). That is, \(\mathfrak{g}=\mathfrak{g}_0\otimes_\mathbb{R}\mathbb{C}\).
- The Killing form is negative definite on \(\mathfrak{g}_0\).
Every complex semisimple Lie group has a compact form. For example, the compact form of \(\SL(n,\mathbb{C})\) is \(\SU(n)\), that of \(\SO(n,\mathbb{C})\) is \(\SO(n)\), and that of \(\Sp(2n,\mathbb{C})\) is \(\Sp(n)=\Sp(2n,\mathbb{C})\cap\U(2n)\).
We now state the central result.
Proposition 15 For a compact connected Lie group \(G\), its complexification \(G_\mathbb{C}\), a maximal torus \(T\subset G\), and the corresponding Borel subgroup \(B\subset G_\mathbb{C}\), the inclusion
\[G/T\hookrightarrow G_\mathbb{C}/B\]is a homotopy equivalence. In particular, \(G/T\) and \(G_\mathbb{C}/B\) have the same cohomology.
Proof
Consider the Iwasawa decomposition \(G_\mathbb{C}=G\cdot A\cdot N\) of \(G_\mathbb{C}\), where
- \(A=\exp(i\mathfrak{t})\) is the abelian subgroup corresponding to the split real form of \(T\).
- \(N=\exp(\mathfrak{n})\) is the unipotent subgroup corresponding to \(\mathfrak{n}=\bigoplus_{\alpha>0}\mathfrak{g}_\alpha\).
The Iwasawa decomposition implies \(G\cap (A\cdot N)=\{e\}\) (the decomposition of each element is unique). The Borel subgroup \(B\) contains the complex maximal torus \(T_\mathbb{C}=T\cdot A\) and decomposes as \(B=T\cdot A\cdot N\), so \(G\cap B = T\cdot(G\cap A\cdot N)=T\). Now consider the following chain.
\[G/T\hookrightarrow G_\mathbb{C}/B=(G\cdot A\cdot N)/(T\cdot A\cdot N)\cong G/(G\cap T\cdot A\cdot N)=G/T\]The first inclusion is induced from \(G\hookrightarrow G_\mathbb{C}\), and since the composition is the identity map on \(G/T\), this inclusion is a homotopy equivalence.
More precisely, because \(A\cdot N\cong\mathbb{R}^n\) is contractible (as Euclidean space), the map \(G_\mathbb{C}/B\to G/T\) induces a deformation retraction.
This result means that \(G/T\) from the compact Lie group viewpoint and the flag variety \(G_\mathbb{C}/B\) from the complex Lie group viewpoint are essentially the same object. In particular, one can use the algebro-geometric properties of the flag variety to study the topological properties of \(G/T\) — its cohomology, homotopy groups, and so on.
Bruhat Decomposition
Finally we introduce an important decomposition of \(G_\mathbb{C}\). This decomposition is essential for understanding the cell structure of the flag variety.
Proposition 16 For a complex semisimple Lie group \(G_\mathbb{C}\), a Borel subgroup \(B\), and the Weyl group \(W\), the following decomposition holds.
\[G_\mathbb{C}=\bigsqcup_{w\in W}BwB\]This is called the Bruhat decomposition. A more detailed discussion of this and its generalization to parabolic subgroups is given in §Bruhat Decomposition and Parabolic Subgroups, ⁋Theorem 4. Each double coset \(BwB\) is a locally closed subset of \(G_\mathbb{C}\), and its closure is given by
\[\overline{BwB}=\bigcup_{v\leq w}BvB\]where \(\leq\) is the Bruhat order on the Weyl group.
Proof
First we show that \(G_\mathbb{C}=\bigcup_{w\in W}BwB\). For any \(g\in G_\mathbb{C}\), one checks whether \(g^{-1}B\cap T\neq\emptyset\), and then uses elements of the Weyl group to send \(g\) into the appropriate double coset.
To see disjointness, suppose \(BwB=BvB\). Then \(wBw^{-1}=vBv^{-1}\), which implies \(w^{-1}v\in N(T)\) and that \(w^{-1}v\) normalizes \(B\). But since \(B\cap N(T)=T\), we have \(w^{-1}v\in T\), and therefore \(w=v\) in \(W\).
The statement about closure follows from the definition of the Bruhat order.
The Bruhat decomposition provides a cell decomposition of the flag variety \(G_\mathbb{C}/B\). For each \(w\in W\), the Schubert cell \(X_w=BwB/B\) is isomorphic to an affine space of dimension \(\ell(w)\), and together they cover all of \(G_\mathbb{C}/B\). Here \(\ell(w)\) is the length of \(w\), i.e. the minimal number of simple reflections needed to express \(w\).
Example 17 When \(G_\mathbb{C}=\GL(n,\mathbb{C})\), the Weyl group is \(W\cong S_n\), and for each permutation \(\sigma\in S_n\), the length \(\ell(\sigma)\) is the number of inversions. The concrete form of this decomposition and its relation to Gaussian elimination are discussed in §Bruhat Decomposition and Parabolic Subgroups, ⁋Example 6.
Specifically, an inversion of \(\sigma\) is a pair \((i,j)\) with \(i<j\) and \(\sigma(i)>\sigma(j)\). By the Bruhat decomposition, \(\GL(n,\mathbb{C})/B\) has a cell decomposition ranging from a \(0\)-dimensional cell (the identity permutation, with \(0\) inversions) up to an \(n(n-1)/2\)-dimensional cell (the reverse permutation, with the maximum number of inversions).
From this cell decomposition one can compute the cohomology of \(\GL(n,\mathbb{C})/B\), and its Betti numbers are determined by the Bruhat order on the Weyl group.
References
[BtD] Theodor Bröcker, Tammo tom Dieck, Representations of Compact Lie Groups, Graduate texts in mathematics, Springer, 1985.
[Hum] James E. Humphreys, Linear Algebraic Groups, Graduate texts in mathematics, Springer, 1975.
[Spr] T. A. Springer, Linear Algebraic Groups, Progress in mathematics, Birkhäuser, 1998.
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