🔁Lie Algebras and Lie Groups Unit 9 – Lie Group Actions & Homogeneous Spaces

Key Concepts and Definitions

• Lie group $G$ smooth manifold equipped with group structure where multiplication and inversion are smooth maps
• Lie algebra $\mathfrak{g}$ tangent space of a Lie group at the identity element, equipped with a bracket operation
• Group action map $\phi: G \times X \to X$ satisfying identity and compatibility conditions, where $G$ is a group and $X$ is a set
  • Left action $g \cdot (h \cdot x) = (gh) \cdot x$
  • Right action $(x \cdot g) \cdot h = x \cdot (gh)$
• Orbit $\mathcal{O}_x$ set of elements in $X$ obtained by applying group elements to $x \in X$, i.e., $\mathcal{O}_x = \{g \cdot x \mid g \in G\}$
• Stabilizer subgroup $G_x$ set of group elements that fix a point $x \in X$, i.e., $G_x = \{g \in G \mid g \cdot x = x\}$
• Homogeneous space $X$ transitive $G$-space, i.e., for any $x, y \in X$, there exists a $g \in G$ such that $g \cdot x = y$
• Coset space $G/H$ set of left cosets $\{gH \mid g \in G\}$ of a subgroup $H$ in $G$

Group Actions and Their Properties

• Group action $\phi$ associates each element of a group $G$ with a transformation of a set $X$ while preserving the group structure
• Identity element $e \in G$ acts as the identity transformation on $X$, i.e., $e \cdot x = x$ for all $x \in X$
• Compatibility condition $(gh) \cdot x = g \cdot (h \cdot x)$ ensures consistency between group multiplication and action
• Faithful action injective homomorphism from the group to the symmetry group of $X$, i.e., different group elements induce different transformations
• Transitive action for any two elements $x, y \in X$, there exists a group element $g \in G$ such that $g \cdot x = y$
  • Enables the study of $X$ through the group structure of $G$
• Free action stabilizer of each point is trivial, i.e., $G_x = \{e\}$ for all $x \in X$
  • Each group element moves every point, except for the identity

Orbits and Stabilizers

• Orbit $\mathcal{O}_x$ set of elements in $X$ reachable from $x$ by applying group actions, i.e., $\mathcal{O}_x = \{g \cdot x \mid g \in G\}$
  • Orbits partition the set $X$ into disjoint equivalence classes
  • Example: rotations of a sphere; each orbit is a circle of latitude
• Stabilizer $G_x$ subgroup of $G$ that fixes the point $x$, i.e., $G_x = \{g \in G \mid g \cdot x = x\}$
  • Measures the symmetry of the action at the point $x$
  • Larger stabilizer implies more symmetry at $x$
• Orbit-stabilizer theorem $|G| = |\mathcal{O}_x| \cdot |G_x|$ relates the size of the group, orbit, and stabilizer
  • Provides a way to compute the size of an orbit using the stabilizer
• Conjugacy class set of elements in $G$ conjugate to each other, i.e., $\{ghg^{-1} \mid g \in G\}$ for some $h \in G$
  • Orbits of the conjugation action of $G$ on itself

Homogeneous Spaces: Structure and Examples

• Homogeneous space $X$ transitive $G$-space, i.e., for any $x, y \in X$, there exists a $g \in G$ such that $g \cdot x = y$
  • Group action allows for "moving" between any two points in the space
• Coset space $G/H$ natural homogeneous space arising from a subgroup $H$ of $G$
  • Elements of $G/H$ are left cosets $gH = \{gh \mid h \in H\}$
  • $G$ acts transitively on $G/H$ by left multiplication
• Isotropy subgroup $G_x$ stabilizer of a point $x \in X$, i.e., $G_x = \{g \in G \mid g \cdot x = x\}$
  • All isotropy subgroups are conjugate for a transitive action
• Examples:
  • Spheres $S^n$ homogeneous spaces under the action of orthogonal groups $O(n+1)$
  • Projective spaces $\mathbb{RP}^n, \mathbb{CP}^n$ homogeneous spaces under the action of projective linear groups $PGL(n+1, \mathbb{R}), PGL(n+1, \mathbb{C})$
  • Grassmannians $Gr(k, n)$ homogeneous spaces under the action of $GL(n, \mathbb{R})$ or $GL(n, \mathbb{C})$

Transitive and Free Actions

• Transitive action for any $x, y \in X$, there exists a $g \in G$ such that $g \cdot x = y$
  • Orbits coincide with the entire space $X$
  • Allows for studying $X$ through the group structure of $G$
• Free action stabilizer of each point is trivial, i.e., $G_x = \{e\}$ for all $x \in X$
  • Each group element moves every point, except for the identity
  • Orbits are in one-to-one correspondence with the group $G$
• Proper action continuous map $G \times X \to X \times X$ given by $(g, x) \mapsto (x, g \cdot x)$ is proper
  • Generalizes the notion of a free action
  • Ensures the existence of slices and a well-behaved quotient space $X/G$
• Examples:
  • $SO(3)$ acting transitively on the sphere $S^2$
  • $\mathbb{R}$ acting freely on itself by translation
  • $GL(n, \mathbb{R})$ acting properly on the space of invertible matrices

Quotient Spaces and Coset Spaces

• Quotient space $X/G$ set of orbits $\{\mathcal{O}_x \mid x \in X\}$ under the action of $G$ on $X$
  • Each point in $X/G$ represents an entire orbit in $X$
  • Inherits topological and geometric properties from $X$ and $G$
• Coset space $G/H$ set of left cosets $\{gH \mid g \in G\}$ of a subgroup $H$ in $G$
  • Natural quotient space arising from the action of $H$ on $G$ by right multiplication
  • Inherits a smooth manifold structure from $G$ when $H$ is a closed subgroup
• Projection map $\pi: X \to X/G$ sends each point $x \in X$ to its orbit $\mathcal{O}_x$
  • Continuous surjection that identifies points in the same orbit
• Examples:
  • Torus $T^2$ quotient of the plane $\mathbb{R}^2$ by the action of $\mathbb{Z}^2$ by translation
  • Real projective space $\mathbb{RP}^n$ quotient of the sphere $S^n$ by the antipodal action of $\mathbb{Z}/2\mathbb{Z}$
  • Grassmannian $Gr(k, n)$ quotient of the Stiefel manifold $V_k(\mathbb{R}^n)$ by the right action of $O(k)$

Applications in Physics and Geometry

• Symmetry groups in physics describe the invariance of physical laws under transformations
  • Examples: Lorentz group in special relativity, Poincaré group in quantum field theory
• Homogeneous spaces as configuration spaces for mechanical systems
  • Example: rigid body motion described by the rotation group $SO(3)$ acting on the sphere $S^2$
• Gauge theories principal bundles with structure group acting on the fibers
  • Connection describes the parallel transport of fibers along curves in the base space
• Riemannian symmetric spaces homogeneous spaces with an additional symmetry condition
  • Sectional curvature is invariant under the action of the symmetry group
  • Examples: Euclidean spaces, spheres, hyperbolic spaces
• Representation theory studying abstract groups through their linear actions on vector spaces
  • Provides a way to analyze the structure and properties of groups using linear algebra techniques

Problem-Solving Techniques and Examples

• Identify the group $G$, the set $X$, and the action $\phi: G \times X \to X$
  • Check that the identity and compatibility conditions are satisfied
• Determine the orbits $\mathcal{O}_x$ and stabilizers $G_x$ for various points $x \in X$
  • Use the orbit-stabilizer theorem to compute the size of orbits
• Classify the action as transitive, free, or proper
  • Transitivity: show that any two points can be connected by a group element
  • Freeness: show that the stabilizer of each point is trivial
  • Properness: show that the map $(g, x) \mapsto (x, g \cdot x)$ is proper
• Construct the quotient space $X/G$ and study its properties
  • Identify the orbits and define the projection map $\pi: X \to X/G$
  • Determine the topology and geometry inherited from $X$ and $G$
• Examples:
  • $\mathbb{Z}$ acting on $\mathbb{R}$ by translation: orbits are cosets, quotient is the circle $S^1$
  • $O(n)$ acting on $\mathbb{R}^n$ by matrix multiplication: orbits are spheres, quotient is the half-line $[0, \infty)$
  • $SL(2, \mathbb{R})$ acting on the upper half-plane $\mathbb{H}$ by Möbius transformations: action is transitive and free, quotient is the moduli space of hyperbolic structures on a torus