Lie Groups and Group Actions

A Lie group is a group that is also a smooth manifold, with smooth multiplication and inversion: a space of symmetries in which one can move continuously. The rotations of space, the rigid motions a robot arm or a spacecraft can make, the unit quaternions, the invertible matrices: all are Lie groups. They matter to the Ricci flow in two ways. First, the most symmetric metrics are those on Lie groups that are invariant under the group's own translations, and five of Thurston's eight model geometries arise this way; for these metrics the Ricci flow, a PDE, collapses to a system of ODEs (11A.8 Homogeneous Flows). Second, Lie groups act on manifolds, and quotients by such actions, such as $S^3/\Gamma$, are among the manifolds the flow must recognise (7A.6 Covering Spaces).

The central fact about a Lie group is that its structure near the identity determines a great deal: the tangent space at the identity, with an operation called the bracket, is the Lie algebra, and the exponential map carries it onto (a neighbourhood of the identity in) the group. For matrix groups, the exponential is the matrix exponential of 2B.6 Power Series, Exponentials and Bump Functions.

By the end of this chapter you will be able to:

• define Lie groups and recognise the standard examples, including $SO(3)$, $SU(2) \cong S^3$, $SE(3)$ and the Heisenberg group;
• define left-invariant vector fields and the Lie algebra, and compute the bracket for matrix groups;
• use the exponential map, including Rodrigues' formula for rotations;
• explain why every Lie group is parallelisable, so $S^2$ is not a Lie group;
• describe smooth group actions, quotient manifolds and homogeneous spaces.

5.1Rigid motions of a robot

5.2Lie groups

Examples.

• $\mathbb{R}^n$ under addition; the circle $S^1 = \{z \in \mathbb{C} : |z| = 1\}$ under multiplication; the torus $T^n$.
• $GL(n, \mathbb{R})$, open in the space of matrices (8A.1 Smooth Structures); matrix multiplication is polynomial in the entries, and inversion is rational (Cramer's rule).
• Closed subgroups of $GL(n)$ that are submanifolds: $O(n)$, $SO(n)$, $SL(n)$ (8A.4 Submanifolds), and the complex groups $U(n)$, $SU(n)$. In fact every closed subgroup of a Lie group is a Lie group (Cartan's closed subgroup theorem).
• $SU(2)$, the unit quaternions, which is the 3-sphere (7A.6 Covering Spaces). So $S^3$ is a Lie group, and so are $S^1$ and $S^0$; these are the only spheres that are.
• $SE(3)$, as above, realised as the $4\times4$ matrices $\begin{pmatrix}R & t\\ 0 & 1\end{pmatrix}$.
• The Heisenberg group $\mathrm{Nil}$ of matrices $\begin{pmatrix}1 & x & z\\ 0 & 1 & y\\ 0 & 0 & 1\end{pmatrix}$, diffeomorphic to $\mathbb{R}^3$ but not commutative; and the group $\mathrm{Sol}$ of $\mathbb{R}^3$ with the product $(x, y, z)(x', y', z') = (x + e^zx', y + e^{-z}y', z + z')$. Both carry model geometries of Thurston (10A.5 Thurston’s Eight Geometries).

5.3Left-invariant vector fields and the Lie algebra

Each $g \in G$ gives a diffeomorphism $L_g(h) = gh$, left translation. A vector field $X$ on $G$ is left-invariant if $dL_g(X_h) = X_{gh}$ for all $g$, $h$: it looks the same at every point, transported by the group. A left-invariant field is determined by its value at the identity $e$, $X_g = dL_g(X_e)$, and every vector in $T_eG$ extends to one. So left-invariant vector fields form a vector space isomorphic to $T_eG$.

Vector fields have a bracket, $[X, Y]f = X(Yf) - Y(Xf)$, which is again a vector field (8A.6 Flows and the Lie Derivative), and the bracket of left-invariant fields is left-invariant.

For a matrix group $G \subseteq GL(n)$, the Lie algebra is a space of matrices and the bracket is the commutator $[A, B] = AB - BA$ (Lee, chapter 8). From 8A.4 Submanifolds:

group	Lie algebra	dimension
$GL(n)$	all $n\times n$ matrices $\mathfrak{gl}(n)$	$n^2$
$SL(n)$	trace zero $\mathfrak{sl}(n)$	$n^2 - 1$
$O(n)$, $SO(n)$	skew-symmetric $\mathfrak{so}(n)$	$\frac{n(n-1)}{2}$
$SU(2)$	skew-Hermitian, trace zero $\mathfrak{su}(2)$	$3$

For $SO(3)$, a skew matrix is determined by a vector $\omega \in \mathbb{R}^3$, $\hat\omega v = \omega\times v$, and the commutator becomes the cross product: $[\hat\omega, \hat\eta] = \widehat{\omega\times\eta}$ (Exercise 5.4). Rotations about different axes don't commute, and the cross product measures how.

Lie groups are parallelisable. Choose a basis $E_1, \dots, E_n$ of $\mathfrak g$; the left-invariant fields they define are linearly independent at every point, so they trivialise $TG \cong G\times\mathfrak g$. In particular a Lie group has nowhere-vanishing vector fields, so $S^2$ is not a Lie group (hairy ball, 7A.7 Smooth Topology), while $S^1$ and $S^3$ are.

5.4The exponential map

For $X \in \mathfrak g$, the integral curve of the left-invariant field $X$ through $e$ is a one-parameter subgroup: a smooth homomorphism $\gamma_X : \mathbb{R} \to G$ with $\gamma_X'(0) = X$. It exists for all time (left-invariant fields are complete, 8A.6 Flows and the Lie Derivative). The exponential map is

For matrix groups it is the matrix exponential $\exp(A) = \sum_k\frac{A^k}{k!}$ (2B.6 Power Series, Exponentials and Bump Functions), and $\gamma_X(t) = \exp(tX)$. Its differential at $0$ is the identity, so $\exp$ is a diffeomorphism from a neighbourhood of $0$ in $\mathfrak g$ onto a neighbourhood of $e$ in $G$ (2B.9 The Inverse and Implicit Function Theorems): exponential coordinates.

Rotations. For $\omega \in \mathbb{R}^3$ with $|\omega| = \theta$ and unit axis $u = \omega/\theta$, Rodrigues' formula gives

the rotation by angle $\theta$ about $u$ (Exercise 5.5). Every rotation is of this form, so $\exp : \mathfrak{so}(3) \to SO(3)$ is surjective. Restricted to the closed ball $|\omega| \leq \pi$ it is onto and injective except that $\omega$ and $-\omega$ give the same rotation when $|\omega| = \pi$. So $SO(3)$ is the closed ball of radius $\pi$ with antipodal points of its boundary identified (Figure 5.1): another picture of $\mathbb{R}P^3$ (7A.6 Covering Spaces). A loop of rotations about a fixed axis, from $0$ to $2\pi$, runs along a diameter from the centre to the boundary, jumps to the antipodal point and comes back: the non-contractible loop of the belt trick.

Not every Lie group has a surjective exponential: in $SL(2, \mathbb{R})$ the matrix $\begin{pmatrix}-1 & 1\\ 0 & -1\end{pmatrix}$ is not an exponential (Exercise 5.6). Compact connected groups always do.

5.5Group actions and quotients

A Lie group $G$ acts smoothly on a manifold $M$ if there is a smooth map $G\times M \to M$, $(g, p) \mapsto g\cdot p$, with $e\cdot p = p$ and $g\cdot(h\cdot p) = (gh)\cdot p$. The action is free if no $g \neq e$ fixes any point, and proper if the map $(g, p) \mapsto (g\cdot p, p)$ is proper (7A.2 Compactness and Compactification); actions of compact groups are always proper.

For a discrete group acting freely and properly (a covering space action, 7A.6 Covering Spaces), $M/G$ is a smooth manifold of the same dimension, and the projection is a local diffeomorphism: this is how lens spaces, $\mathbb{R}P^n$, flat tori and the Poincaré homology sphere become smooth manifolds.

Homogeneous spaces. If $G$ acts transitively on $M$, then $M \cong G/H$, where $H$ is the subgroup fixing a point. The sphere $S^2$ is $SO(3)/SO(2)$: rotations act transitively, and the rotations fixing the north pole are rotations about the vertical axis. Hyperbolic space and every model geometry of Thurston is a homogeneous space (10A.5 Thurston’s Eight Geometries). On a homogeneous space a $G$-invariant metric is determined by an inner product on one tangent space, so the Ricci flow among invariant metrics is an ODE.

5.6History

Sophus Lie developed continuous transformation groups from 1873 onwards to study symmetries of differential equations; Wilhelm Killing and Élie Cartan classified the simple Lie algebras in 1888–94, and Cartan, Hermann Weyl and others developed the global theory of Lie groups in the 1920s and 30s. Olinde Rodrigues published his rotation formula in 1840. Cartan's closed subgroup theorem dates from 1930. The use of Lie groups in robot motion and estimation became widespread from the 1990s on.

5.7Exercises

Solution

$[\hat\omega, \hat\eta]v = \omega\times(\eta\times v) - \eta\times(\omega\times v)$. The Jacobi identity $\omega\times(\eta\times v) + \eta\times(v\times\omega) + v\times(\omega\times\eta) = 0$ gives $\omega\times(\eta\times v) - \eta\times(\omega\times v) = -v\times(\omega\times\eta) = (\omega\times\eta)\times v$.

Solution

$\hat u^2v = u\times(u\times v) = u(u\cdot v) - v$, so $\hat u^2$ is minus the projection onto $u^\perp$, and $\hat u^3 = -\hat u$. Then $\hat u^{2k+1} = (-1)^k\hat u$ and $\hat u^{2k} = (-1)^{k-1}\hat u^2$ for $k \geq 1$, and the series give $\sin\theta$ and $1 - \cos\theta$. On $u$: $\hat uu = 0$, so $u$ is fixed. On $u^\perp$: $\hat u^2 = -I$, so the map is $\cos\theta\,I + \sin\theta\,\hat u$, a rotation by $\theta$ (with $\hat u$ the rotation by $\frac\pi2$).

Solution

(a) $\partial_tL_a^*g = L_a^*\partial_tg = -2L_a^*\operatorname{Ric}(g) = -2\operatorname{Ric}(L_a^*g)$, and $L_a^*g(0) = L_a^*g_0 = g_0$. (b) By uniqueness, $L_a^*g(t) = g(t)$ for all $t$. A left-invariant metric is $g_e$ transported by $dL_a$, so it is determined by the inner product $g_e$ on $T_eG = \mathfrak g$, a symmetric matrix with $\frac{n(n+1)}{2}$ entries.