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Lie groups and homogeneous spaces

In the Fifties Bott applied Morse theory with great success to the topology of Lie groups and homogeneous spaces. In [8] he showed how the diagram of a compact semisimple connected and simply connected group $ G$ determines the integral homology of both the loop space $ \Omega G$ and the flag manifold $ G/T$ , where $ T$ is a maximal torus.

Indeed, Morse theory gives a beautiful CW cell structure on $ G/T$ , up to homotopy equivalence. To explain this, recall that the adjoint action of the group $ G$ on its Lie algebra $ \mathfrak{g}$ restricts to an action of the maximal torus $ T$ on $ \mathfrak{g}$ . As a representation of the torus $ T$ , the Lie algebra $ \mathfrak{g}$ decomposes into a direct sum of irreducible representations

$\displaystyle \mathfrak{g}= \mathfrak{t}\oplus \sum E_{\alpha} ,

where $ \mathfrak{t}$ is the Lie algebra of $ T$ and each $ E_{\alpha }$ is a $ 2$ -dimensional space on which $ T$ acts as a rotation $ e^{2\pi i \alpha
(x)}$ , corresponding to the root $ \alpha (x)$ on $ \mathfrak{t}$ . The diagram of $ G$ is the family of parallel hyperplanes in $ \mathfrak{t}$ where some root is integral. A hyperplane that is the zero set of a root is called a root plane.

For example, for the group $ G = \operatorname{SU}(3)$ and maximal torus

$\displaystyle T=\left\{ \left.
e^{2\pi i x_1} & & \\
& e^{2\p...
\right\vert \
x_1+x_2+x_3 = 0, x_i \in \mathbb{R}\right\} .

the roots are $ \pm(x_1 - x_2), \pm(x_1 - x_3) , \pm(x_2 - x_3)$ , and the diagram is the collection of lines in the plane $ x_1+x_2+x_3=0$ in $ \mathbb{R}^3$ as in Figure 3. In this figure, the root planes are the thickened lines.

Figure 3: The diagram of SU(3)

For $ G=\operatorname{SU}(2)$ and

$\displaystyle T= \left\{ \left.
e^{2\pi i x} & 0 \cr
0 & e^{-2\pi i x}
\end{bmatrix}\ \right\vert \
x \in\mathbb{R}\right\},

the Lie algebra $ \mathfrak{t}$ is $ \mathbb{R}$ , the roots are $ \pm 2x$ , and the adjoint representation of of $ G$ on $ \mathfrak{g}= \mathbb{R}^3$ corresponds to rotations. The root plane is the origin.

Figure 4: The diagram of SU(2)

A point $ B$ in $ \mathfrak{t}$ is regular if its normalizer has minimal possible dimension, or equivalently, if its normalizer is $ T$ . It is well known that a point $ B$ in $ \mathfrak{t}$ is regular if and only if it does not lie on any of the hyperplanes of the diagram. If $ B$ is regular, then the stabilizer of $ B$ under the adjoint action of $ G$ is $ T$ and so the orbit through $ B$ is $ G/T$ .

Choose another regular point $ A$ in $ \mathfrak{t}$ and define the function $ f$ on $ \operatorname{Orbit}(B) = G/T$ to be the distance from $ A$ ; here the distance is measured with respect to the Killing form on $ \mathfrak{g}$ . Let $ \{ B_i\}$ be all the points in $ \mathfrak{t}$ obtained from $ B$ by reflecting about the root planes. Then Bott's theorem asserts that $ f$ is a Morse function on $ G/T$ whose critical points are precisely all the $ B_i$ 's. Moreover, the index of a critical point $ B_i$ is twice the number of times that the line segment from $ A$ to $ B_i$ intersects the root planes. This cell decomposition of Morse theory fits in with the more group-theoretic Bruhat decomposition.

For $ G = \operatorname{SU}(3)$ and $ T$ the set of diagonal matrices in $ \operatorname{SU}(3)$ , the orbit $ G/T$ is the complex flag manifold $ \operatorname {F\ell }(1,2,3)$ , consisting of all flags

$\displaystyle V_1 \subset V_2 \subset \mathbb{C}^3, \quad \dim_{\mathbb{C}} V_i = i.

Bott's recipe gives 6 critical points of index $ 0,2,2,4,4,6$ respectively on $ G/T$ (See Figure 5). By the lacunary principle, the Morse function $ f$ is perfect. Hence, the flag manifold $ \operatorname {F\ell }(1,2,3)$ has the homotopy type of a CW complex with one 0 -cell, two $ 2$ -cells, two $ 4$ -cells, and one $ 6$ -cell. Its Poincaré polynomial is therefore

$\displaystyle P_t(\operatorname{F\ell}(1,2,3)) = 1+2t^2 +2t^4 +t^6.

Figure 5: The flag manifold $ \operatorname {F\ell }(1,2,3)$

next up previous
Next: Index of a closed Up: The life and works Previous: Morse theory
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