9.1 Lie group actions and orbits

Lie group actions are a powerful tool for studying symmetries in mathematics and physics. They describe how a Lie group transforms a manifold, revealing important geometric and physical properties through the concept of orbits and stabilizer subgroups.

This section dives into the definition and properties of Lie group actions, exploring their applications in geometry and physics. We'll examine orbits, stabilizer subgroups, and key theorems that provide deep insights into the structure of these actions.

Lie group actions and applications

Definition and properties of Lie group actions

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• A Lie group action is a smooth map $\phi: G \times M \to M$, where $G$ is a Lie group and $M$ is a smooth manifold, satisfying the following properties:
  1. Identity: $\phi(e, x) = x$ for all $x \in M$, where $e$ is the identity element of $G$
  2. Compatibility: $\phi(g_1, \phi(g_2, x)) = \phi(g_1g_2, x)$ for all $g_1, g_2 \in G$ and $x \in M$
• The action of a Lie group $G$ on a manifold $M$ induces a homomorphism from $G$ to the group of diffeomorphisms of $M$
  • This homomorphism captures the infinitesimal transformations generated by the Lie algebra of $G$

Applications in geometry and physics

• Lie group actions describe symmetries and invariance properties of geometric objects
  • Example: The action of the rotation group $[SO(3)](https://www.fiveableKeyTerm:so(3))$ on the sphere $S^2$ preserves the shape and orientation of the sphere
• In physics, Lie group actions model the symmetries of physical systems
  • Example: The action of the Lorentz group on spacetime in special relativity describes the invariance of physical laws under changes of reference frame
  • Example: The action of the gauge group on the configuration space of a gauge theory captures the internal symmetries of the theory (Yang-Mills theory)

Orbits of Lie group actions

Definition and properties of orbits

• The orbit of a point $x \in M$ under a Lie group action $\phi: G \times M \to M$ is the set $O_x = \{\phi(g, x) | g \in G\}$
  • Orbits consist of all points in $M$ that can be reached from $x$ by applying elements of $G$
• Orbits partition the manifold $M$ into disjoint subsets
  • The set of all orbits is called the orbit space $M/G$
• The dimension of an orbit $O_x$ is equal to the dimension of the Lie group $G$ minus the dimension of the stabilizer subgroup $G_x = \{g \in G | \phi(g, x) = x\}$

Classification of orbits

• Orbits can be classified as stable, unstable, or semistable based on their behavior under small perturbations
  • Stable orbits remain close to their original position
  • Unstable orbits diverge from their original position
• The stability of an orbit can be determined by analyzing the eigenvalues of the linearized action of the Lie group at a point on the orbit
  • Example: In the action of $SO(2)$ on $\mathbb{R}^2$, the origin is a stable fixed point, while other orbits are stable circles

Stabilizer subgroup of a Lie group action

Definition and properties of stabilizer subgroups

• The stabilizer subgroup (or isotropy subgroup) of a point $x \in M$ under a Lie group action $\phi: G \times M \to M$ is the subgroup $G_x = \{g \in G | \phi(g, x) = x\}$
  • $G_x$ consists of all elements of $G$ that leave $x$ fixed
• The stabilizer subgroup $G_x$ is a closed subgroup of $G$ and is itself a Lie group
• The dimension of the stabilizer subgroup $G_x$ is related to the dimension of the orbit $O_x$ through the orbit-stabilizer theorem: $\dim(O_x) + \dim(G_x) = \dim(G)$

Conjugacy of stabilizer subgroups

• The stabilizer subgroups at different points on the same orbit are conjugate to each other
  • If $y = \phi(h, x)$ for some $h \in G$, then $G_y = hG_xh^{-1}$
• Conjugate subgroups have the same algebraic properties and are isomorphic as Lie groups
  • Example: In the action of $SO(3)$ on $S^2$, the stabilizer subgroups at the north and south poles are conjugate to each other and isomorphic to $SO(2)$

Fundamental theorems of Lie group actions

Orbit-stabilizer theorem

• The orbit-stabilizer theorem states that for a Lie group action $\phi: G \times M \to M$ and a point $x \in M$, there is a bijection between the orbit $O_x$ and the quotient space $G/G_x$, where $G_x$ is the stabilizer subgroup of $x$
  • The bijection is given by the map $\psi: G/G_x \to O_x$, defined by $\psi(gG_x) = \phi(g, x)$, which is well-defined and bijective
• The orbit-stabilizer theorem implies that $\dim(O_x) = \dim(G) - \dim(G_x)$
• The proof of the orbit-stabilizer theorem relies on the properties of the Lie group action and the stabilizer subgroup, as well as the smoothness of the maps involved

Other important theorems

• The slice theorem describes the local structure of the orbit space near a point
  • It states that locally, the orbit space is a product of the orbit and a transversal slice
  • The slice theorem is useful for understanding the singularities and stratification of the orbit space
• The principal orbit theorem characterizes the orbits of maximal dimension
  • It states that the set of points whose orbits have maximal dimension is open and dense in the manifold $M$
  • The principal orbit theorem is important for classifying the different types of orbits and understanding the global structure of the action