Borel Subgroups

In §Root Systems, for a semisimple Lie algebra \(\mathfrak{g}\), we defined the root system \(\Phi\) and verified that its symmetries are captured by the Weyl group. In §Torus Actions, we also examined, for a compact Lie group \(G\), the relationship between its maximal torus \(T\) 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. Between two vertices \(\alpha_i\) and \(\alpha_j\) (\(i\neq j\)), place \(\langle\alpha_i,\alpha_j\rangle\langle\alpha_j,\alpha_i\rangle\) edges.
3. If \(\lvert\alpha_i\rvert\neq\lvert\alpha_j\rvert\), add an arrow on the edge pointing toward the shorter root.

In the Cartan matrix \(A=(a_{ij})\), since \(a_{ij}=\langle\alpha_i,\alpha_j\rangle\), one may regard the Dynkin diagram as representing the information of the Cartan matrix as a graph. 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. The simple roots can be chosen as \(\alpha_i=e_i-e_{i+1}\) (\(1\leq i\leq n\)). 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 \(\langle\alpha_i,\alpha_j\rangle\) is, for \(i=j\), \(2\); for \(\lvert i-j\rvert=1\), \(-1\); 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. If we consider, for each \(\Delta_i\), the root subsystem \(\Phi_i\) generated by it, then by Definition 1, 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. That is, a Dynkin diagram is always a tree or a forest.

Proof

Suppose for contradiction that there exists a cycle \(\alpha_{i_1},\ldots,\alpha_{i_k}=\alpha_{i_1}\). The number of edges in the cycle is \(k-1\), and for each edge, \(\langle\alpha_{i_j},\alpha_{i_{j+1}}\rangle\neq 0\), and therefore \(\langle\alpha_{i_j},\alpha_{i_{j+1}}\rangle\leq -1\).

Now for each \(j\), let \(e_j=\alpha_{i_j}/\lvert\alpha_{i_j}\rvert\) and consider \(e=\sum_{j=1}^{k-1}e_j\). Then

\[(e,e)=\sum_{j=1}^{k-1}(e_j,e_j)+2\sum_{j<\ell}(e_j,e_\ell)\]

Here \((e_j,e_j)=1\), and for \(e_j,e_\ell\) connected by an edge, \((e_j,e_\ell)<0\) and \(4(e_j,e_\ell)^2=\langle\alpha_{i_j},\alpha_{i_\ell}\rangle\langle\alpha_{i_\ell},\alpha_{i_j}\rangle\geq 1\), so \(2(e_j,e_\ell)\leq -1\). Since it is a cycle, each \(\alpha_{i_j}\) is connected to exactly two neighbors, and therefore

\[(e,e)\leq (k-1)-(k-1)=0\]

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 is strictly less than \(4\). That is, for any simple root \(\alpha\),

\[\sum_{\beta\in\Delta,\beta\neq\alpha}\langle\alpha,\beta\rangle\langle\beta,\alpha\rangle<4\]

Proof

Consider the simple roots \(\beta_1,\ldots,\beta_m\) adjacent to \(\alpha\). If distinct \(\beta_i\) and \(\beta_j\) were connected by an edge, then \(\alpha,\beta_i,\beta_j\) would form a cycle, so by Proposition 4, the \(\beta_i\) are mutually orthogonal.

Now let \(e_0=\alpha/\lvert\alpha\rvert\) and \(e_i=\beta_i/\lvert\beta_i\rvert\). Then \(e_1,\ldots,e_m\) are orthonormal, and by the linear independence of the simple roots, \(e_0\) does not belong to their span, so by Bessel’s inequality we have \(\sum_{i=1}^m(e_0,e_i)^2<(e_0,e_0)=1\). However, since \(\langle\alpha,\beta_i\rangle\langle\beta_i,\alpha\rangle=4(e_0,e_i)^2\), the total sum is strictly less than \(4\).

ADE Classification

Let us now examine the classification of irreducible root systems. From the preceding propositions, the possible shapes of Dynkin diagrams are significantly restricted.

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
   • \(B_n\) (\(n\geq 2\)): one vertex attached to one end of \(A_{n-1}\) with a double edge, and an arrow added
   • \(C_n\) (\(n\geq 3\)): one vertex attached to one end of \(A_{n-1}\) with a double edge, and an arrow added in the opposite direction
   • \(D_n\) (\(n\geq 4\)): branching at one end of \(A_{n-1}\)
2. Exceptional types:
   • \(E_6, E_7, E_8\): special forms having 6, 7, and 8 vertices, respectively
   • \(F_4\): 4 vertices, with a double edge in the middle
   • \(G_2\): 2 vertices, triple edge

The proof of this classification consists of systematically analyzing the conditions that a Dynkin diagram must satisfy. The key ideas are as follows.

1. Since there are no cycles, it must be in the form of a tree.
2. Since branching is restricted, the possible forms are limited.
3. From constraints on the number of edges, the positions of double edges and triple edges are determined.

We omit the detailed proof, but what is important for understanding the content of the theorem is knowing what features each type has.

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 \(E_6, E_7, E_8, F_4, G_2\) are not 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 have special properties.

Definition 8 A root system \(\Phi\) is simply-laced if \(\lvert\alpha\rvert=\lvert\beta\rvert\) for all \(\alpha,\beta\in\Phi\). When \(\Phi\) is irreducible, this is equivalent to the Dynkin diagram having no double edge or triple edge.

The simply-laced root systems are precisely types \(A_n\), \(D_n\), \(E_6\), \(E_7\), and \(E_8\), and these are collectively called ADE type. The ADE type appears in various mathematical contexts: for example, ADE patterns appear in the classification of du Val singularities, the symmetry groups of Platonic solids, and two-dimensional conformal field theory.

Their Dynkin diagrams are as follows. All of them consist of single edges only, with \(A_n\) being a chain, \(D_n\) a fork branching at one end, and \(E_6,E_7,E_8\) having three branches meeting at a single vertex.

Borel Subalgebra

Now let us examine subalgebras of a Lie algebra that are naturally defined from a root system. In §Root Systems, ⁋Definition 15, we defined the collection \(\Phi^+\) of positive roots. This is equivalent 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 collection of positive roots \(\Phi^+\), the Borel subalgebra is the following subalgebra.

\[\mathfrak{b}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Phi^+}\mathfrak{g}_\alpha\]

In this case, \(\mathfrak{n}=\bigoplus_{\alpha>0}\mathfrak{g}_\alpha\) 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.

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

(2) Suppose \(\mathfrak{b}\) is contained in a solvable subalgebra \(\mathfrak{s}\). 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 any negative root, then \(\mathfrak{s}\subseteq\mathfrak{b}\), and if \(S\) contains a negative root \(-\alpha\), then \(\mathfrak{g}_\alpha,\mathfrak{g}_{-\alpha},[\mathfrak{g}_\alpha,\mathfrak{g}_{-\alpha}]\) forms, inside \(\mathfrak{s}\), a copy of \(\mathfrak{sl}_2\), and since \(\mathfrak{sl}_2\) is simple, \(\mathfrak{s}\) 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 a Borel subalgebra for some positive system.

Borel Subgroup and Flag Variety

Now let us turn to the Lie group perspective. For a complex semisimple Lie group \(G_\mathbb{C}\), we can consider the Lie subgroup corresponding to the Borel subalgebra \(\mathfrak{b}\).

Definition 11 For a complex semisimple Lie group \(G_\mathbb{C}\), the Borel subgroup \(B\) is the connected Lie subgroup corresponding to the 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 was given because in the case of \(\GL(n;\mathbb{C})\), \(\mathcal{F}\) coincides with the space of complete flags in \(\mathbb{C}^n\). In general, the flag variety is a projective variety, and this has a deep connection 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 collection of upper triangular matrices, and the flag variety \(\GL(n;\mathbb{C})/B\) is in bijection with the space of complete flags in \(\mathbb{C}^n\),

\[0=V_0\subseteq V_1\subseteq V_2\subseteq\cdots\subseteq V_n=\mathbb{C}^n,\qquad \dim V_i=i\]

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

We now 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 a compact group and \(G_\mathbb{C}/B\) for a 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.

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 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 \(\Sp(n)=\Sp(2n;\mathbb{C})\cap\Umat(2n)\).

We now state the key result.

Proposition 15 For a compact connected semisimple Lie group \(G\), its complexification \(G_\mathbb{C}\), a maximal torus \(T\subseteq G\), and the corresponding Borel subgroup \(B\subseteq G_\mathbb{C}\), 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 of \(G_\mathbb{C}\), \(G_\mathbb{C}=G\cdot A\cdot N\), where

• \(A=\exp(i\mathfrak{t})\) is the abelian subgroup corresponding to the split real form of \(T\).
• \(N=\exp(\mathfrak{n})\) 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). Because the Borel subgroup \(B\) contains the complex maximal torus \(T_\mathbb{C}=T\cdot A\) and decomposes as \(B=T\cdot A\cdot N\), we have \(G\cap B = T\cdot(G\cap A\cdot N)=T\). 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 by \(G\hookrightarrow G_\mathbb{C}\); since the composition is the identity map on \(G/T\), this inclusion is a homotopy equivalence.

More precisely, since \(A\cdot N\cong\mathbb{R}^n\) (contractible as a Euclidean space), \(G_\mathbb{C}/B\rightarrow G/T\) induces a deformation retraction.

This result means that \(G/T\) from the viewpoint of compact Lie groups and the flag variety \(G_\mathbb{C}/B\) from the viewpoint of complex Lie groups are essentially the same object. In particular, one can exploit the algebro-geometric properties of the flag variety to study the topological properties of \(G/T\) (cohomology, homotopy groups, etc.).

Bruhat Decomposition

Finally, we introduce an important decomposition of \(G_\mathbb{C}\). This decomposition is crucial 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 are treated in §Bruhat Decomposition, ⁋Theorem 5 (Bruhat decomposition). Each double coset \(BwB\) is a locally closed subset of \(G_\mathbb{C}\), and its closure is given as follows:

\[\overline{BwB}=\bigcup_{v\leq w}BvB\]

Here \(\leq\) is the Bruhat order on the Weyl group.

Proof

First, let us show that \(G_\mathbb{C}=\bigcup_{w\in W}BwB\). For any \(g\in G_\mathbb{C}\), the two Borel subgroups \(B\) and \(gBg^{-1}\) contain a common maximal torus, and conjugating it to \(T\) inside \(B\), there is some \(b\in B\) such that \(bgBg^{-1}b^{-1}\) contains \(T\). Since a Borel subgroup containing \(T\) is of the form \(wBw^{-1}\) (\(w\in W\)), we have \(bgBg^{-1}b^{-1}=wBw^{-1}\), and since the normalizer of \(B\) is \(B\) itself, \(w^{-1}bg\in B\), that is, \(g\in BwB\).

To see disjointness, suppose \(BwB=BvB\). Then we can write \(v=b_1wb_2\) (\(b_1,b_2\in B\)), so \(vBv^{-1}=b_1(wBw^{-1})b_1^{-1}\). Now \(T\) and \(b_1^{-1}Tb_1\) are both maximal tori contained in the solvable group \(B\cap wBw^{-1}\), so for some \(c\in B\cap wBw^{-1}\) we have \(cb_1^{-1}Tb_1c^{-1}=T\), and then \(b_1c^{-1}\in B\cap N(T)=T\), which implies \(b_1\in wBw^{-1}\). Therefore \(wBw^{-1}=vBv^{-1}\), from which \(w^{-1}v\in N(T)\) 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 the 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 \(w\in W\), \(X_w^\circ=BwB/B\) is isomorphic to an affine space of dimension \(\ell(w)\), and collecting these covers all of \(G_\mathbb{C}/B\). Here \(\ell(w)\) is the length of \(w\), that is, the minimum number needed when expressing \(w\) as a product of simple reflections.

Example 17 In the case where \(G_\mathbb{C}=\GL(n;\mathbb{C})\), the Weyl group is \(W\cong S_n\), and for each permutation \(\sigma\in S_n\), \(\ell(\sigma)\) is the number of inversions. The concrete form of this decomposition and its relation to Gauss elimination are discussed in §Bruhat Decomposition, ⁋Example 7.

Specifically, an inversion of \(\sigma\) is, with \(i<j\) and \(\sigma(i)>\sigma(j)\), the number of pairs \((i,j)\). By the Bruhat decomposition, \(\GL(n;\mathbb{C})/B\) has a cell decomposition ranging from a \(0\)-dimensional cell (identity permutation, \(0\) inversions) to an \(n(n-1)/2\)-dimensional cell (reverse permutation, maximum 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.

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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.