Borel Subgroups

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — 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 symmetries are captured by the Weyl group. In §Torus Actions we also examined the relationship between a 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 its simple roots. The Cartan matrix defined in §Root Systems, ⁋Definition 16 encodes these relations in matrix form, but the structure of a root system can be grasped 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.

1. Place one vertex for each simple root \(\alpha_i\).
2. Place \(\lvert\langle\alpha_i,\alpha_j\rangle\rvert\) edges between the two vertices \(\alpha_i\) and \(\alpha_j\) (\(i\neq j\)).
3. If \(\lvert\alpha_i\rvert\neq\lvert\alpha_j\rvert\), add an arrow on the edge pointing toward the longer root.

Since \(a_{ij}=\langle\alpha_i,\alpha_j\rangle\) in the Cartan matrix \(A=(a_{ij})\), one may regard the Dynkin diagram as a graphical encoding of the Cartan matrix. As we saw in §Root Systems, \(a_{ij}\leq 0\), and \(a_{ij}=0\) is equivalent to \(a_{ji}=0\); hence 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 their inner products gives

\[(\alpha_i,\alpha_j)=\begin{cases}2 & i=j\\ -1 & \lvert i-j\rvert=1\\ 0 & \text{otherwise}\end{cases}\]

Thus \(\langle\alpha_i,\alpha_j\rangle\) equals \(2\) when \(i=j\), \(-1\) when \(\lvert i-j\rvert=1\), and \(0\) otherwise. Hence 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. Let \(\Phi_i\) be the root subsystem generated by each \(\Delta_i\). By §Root Systems, ⁋Proposition 6, roots belonging to distinct \(\Delta_i\) 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 preceding argument would make it reducible, a contradiction.

Proposition 4 A Dynkin diagram contains no cycles; equivalently, every Dynkin diagram is 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 set \(\alpha=\sum_{j=1}^{k-1}\alpha_{i_j}\). 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 the vertices form a cycle, each \(\alpha_{i_j}\) is adjacent 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 incident to any 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 \(\alpha\), and let \(\beta_1,\ldots,\beta_m\) be the simple roots adjacent to \(\alpha\). Each \(\beta_i\) makes a distinct angle with \(H_\alpha\).

Linear independence of the projections of the \(\beta_i\) onto \(H_\alpha\) implies \(m\leq 3\). Moreover, \(\lvert\langle\alpha,\beta_i\rangle\rvert\leq 3\) for each \(\beta_i\), so the total sum does not exceed \(4\).

ADE Classification

Let us now 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.

1. Classical types:
   • \(A_n\) (\(n\geq 1\)): \(n\) vertices connected in a chain
   • \(B_n\) (\(n\geq 2\)): a double edge and arrow attached to one end of \(A_n\)
   • \(C_n\) (\(n\geq 2\)): a double edge and arrow in the opposite direction attached to one end of \(A_n\)
   • \(D_n\) (\(n\geq 4\)): a branch at one end of \(A_{n-1}\)
2. 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.

1. Absence of cycles forces the diagram to be a tree.
2. Restricted branching limits the possible forms.
3. Constraints on the number of edges determine where double and triple edges may occur.

We omit the detailed proof; what matters for understanding the theorem is knowing the distinctive features of each type.

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 Systems

A root system in which all roots have the same length is called simply-laced. Such root systems 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 contains 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 wide variety of mathematical contexts: for instance, the classification of du Val singularities, the symmetry groups of Platonic solids, and two-dimensional conformal field theory all exhibit ADE patterns.

Borel Subalgebra

We now turn to subalgebras of a Lie algebra that are defined naturally from a root system. In §Root Systems, ⁋Definition 15 we defined the set \(\Phi^+\) of positive roots. Choosing \(\Phi^+\) amounts to selecting a Weyl chamber, and from this choice we can define a distinguished subalgebra.

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 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 may be regarded as the largest “upper triangular” subalgebra containing the Cartan subalgebra.

Proposition 10 For a Borel subalgebra \(\mathfrak{b}\), the following hold.

1. \(\mathfrak{b}\) is solvable.
2. \(\mathfrak{b}\) is a maximal solvable subalgebra of \(\mathfrak{g}\).
3. Every conjugate of \(\mathfrak{b}\) is again a Borel subalgebra. That is, for any \(g\in G\), \(\Ad(g)\mathfrak{b}\) is the Borel subalgebra associated to 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. Concretely, \(\mathfrak{n}\) has a structure analogous to the algebra of strictly upper triangular matrices.

(2) Suppose \(\mathfrak{s}\) is a solvable subalgebra 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 \(S\) omits some positive root then \(\mathfrak{s}\subset\mathfrak{b}\); if \(S\) contains a negative root then \(\mathfrak{s}\) is no longer solvable. Hence \(\mathfrak{s}=\mathfrak{b}\).

(3) Since \(\Ad(g)\mathfrak{b}\) is again a maximal solvable subalgebra, by (2) it is the Borel subalgebra for some positive system.

Borel Subgroup and Flag Variety

We now adopt the Lie group perspective. For a complex semisimple Lie group \(G_\mathbb{C}\), we may 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 \(G_\mathbb{C}\). We now define the associated quotient space.

Definition 12 For a complex semisimple Lie group \(G_\mathbb{C}\) and a Borel subgroup \(B\), the flag variety is the homogeneous space

\[\mathcal{F}=G_\mathbb{C}/B.\]

The term flag variety arises because for \(\GL(n,\mathbb{C})\), the space \(\mathcal{F}\) coincides with the space of complete flags in \(\mathbb{C}^n\). In general the flag variety is a projective variety, and it is deeply connected to 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 group 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, the coset \(gB\in\GL(n,\mathbb{C})/B\) 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

We now examine the relationship between a compact Lie group \(G\) and its complexification \(G_\mathbb{C}\). This connection furnishes a bridge between the two perspectives: \(G/T\) for compact groups and \(G_\mathbb{C}/B\) for complex groups.

Definition 14 A compact form of a complex Lie group \(G_\mathbb{C}\) is a compact Lie group \(G\) satisfying the following conditions.

1. \(G\) is a Lie subgroup of \(G_\mathbb{C}\).
2. The Lie algebra \(\mathfrak{g}_0\) of \(G\) is a real form of \(\mathfrak{g}\); that is, \(\mathfrak{g}=\mathfrak{g}_0\otimes_\mathbb{R}\mathbb{C}\).
3. The Killing form is negative definite on \(\mathfrak{g}_0\).

Every complex semisimple Lie group admits 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 \(\Sp(n)=\Sp(2n,\mathbb{C})\cap\operatorname{U}(2n)\).

We now state the central result.

Proposition 15 Let \(G\) be a compact connected Lie group, \(G_\mathbb{C}\) its complexification, \(T\subset G\) a maximal torus, and \(B\subset G_\mathbb{C}\) the corresponding Borel subgroup. Then 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 \(G_\mathbb{C}=G\cdot A\cdot N\), where

• \(A=\exp(i\mathfrak{t})\) is the abelian subgroup corresponding to the split real form of \(T\), and
• \(N=\exp(\mathfrak{n})\) 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 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 by \(G\hookrightarrow G_\mathbb{C}\); 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 (being Euclidean space), the projection \(G_\mathbb{C}/B\to G/T\) 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 may study the topological properties of \(G/T\)—its cohomology, homotopy groups, and so on—by exploiting the algebro-geometric properties of the flag variety.

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 Let \(G_\mathbb{C}\) be a complex semisimple Lie group, \(B\) a Borel subgroup, and \(W\) the Weyl group. Then the following decomposition holds:

\[G_\mathbb{C}=\bigsqcup_{w\in W}BwB.\]

This is called the Bruhat decomposition. A more detailed discussion, together with its generalization to parabolic subgroups, is given in §Bruhat Decomposition. 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 \(\leq\) denotes 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 that \(g^{-1}B\cap T\neq\emptyset\), and then uses elements of the Weyl group to move \(g\) into the appropriate double coset.

For disjointness, suppose \(BwB=BvB\). Then \(wBw^{-1}=vBv^{-1}\), whence \(w^{-1}v\in N(T)\) and \(w^{-1}v\) normalizes \(B\). But since \(B\cap N(T)=T\), we have \(w^{-1}v\in T\), and therefore \(w=v\) in \(W\).

The statement concerning closures 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 these cells together 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 \(\ell(\sigma)\) equals the number of inversions. The concrete form of this decomposition and its relation to Gaussian elimination are discussed in §Bruhat Decomposition.

Specifically, an inversion of \(\sigma\) is a pair \((i,j)\) with \(i<j\) and \(\sigma(i)>\sigma(j)\). By the Bruhat decomposition, \(\GL(n,\mathbb{C})/B\) admits 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 \(\GL(n,\mathbb{C})/B\); its Betti numbers are determined by the Bruhat order on the Weyl group.

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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.