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 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 is a smooth map ϕ:G×M→M, where G is a Lie group and M is a smooth manifold, satisfying the following properties:
Identity: ϕ(e,x)=x for all x∈M, where e is the identity element of G
Compatibility: ϕ(g1,ϕ(g2,x))=ϕ(g1g2,x) for all g1,g2∈G and x∈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 S2 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 of a point x∈M under a Lie group action ϕ:G×M→M is the set Ox={ϕ(g,x)∣g∈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 M/G
The dimension of an orbit Ox is equal to the dimension of the Lie group G minus the dimension of the stabilizer subgroup Gx={g∈G∣ϕ(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 R2, the origin is a stable , 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∈M under a Lie group action ϕ:G×M→M is the subgroup Gx={g∈G∣ϕ(g,x)=x}
Gx consists of all elements of G that leave x fixed
The stabilizer subgroup Gx is a closed subgroup of G and is itself a Lie group
The dimension of the stabilizer subgroup Gx is related to the dimension of the orbit Ox through the : dim(Ox)+dim(Gx)=dim(G)
Conjugacy of stabilizer subgroups
The stabilizer subgroups at different points on the same orbit are conjugate to each other
If y=ϕ(h,x) for some h∈G, then Gy=hGxh−1
Conjugate subgroups have the same algebraic properties and are isomorphic as Lie groups
Example: In the action of SO(3) on S2, 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 ϕ:G×M→M and a point x∈M, there is a bijection between the orbit Ox and the G/Gx, where Gx is the stabilizer subgroup of x
The bijection is given by the map ψ:G/Gx→Ox, defined by ψ(gGx)=ϕ(g,x), which is well-defined and bijective
The orbit-stabilizer theorem implies that dim(Ox)=dim(G)−dim(Gx)
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
Key Terms to Review (18)
Borel's Fixed Point Theorem: Borel's Fixed Point Theorem states that any continuous map from a compact convex subset of a Euclidean space into itself has at least one fixed point. This theorem is significant in the study of fixed points, as it provides a foundational result in topology and analysis, particularly in understanding the behavior of continuous functions within bounded regions.
Closed Orbit: A closed orbit is a type of orbit in which a point in a space returns to its initial position after a certain period under the action of a Lie group. This concept is crucial in understanding how Lie groups can act on manifolds, leading to structures that are both geometrically rich and algebraically significant. Closed orbits often indicate that the dynamics of the system exhibit periodic behavior, allowing for deeper insights into the underlying symmetries and conservation laws.
Coadjoint orbit: A coadjoint orbit is a concept in the theory of Lie groups that refers to the orbit of a linear functional under the action of a Lie group via the coadjoint representation. Specifically, it describes how a functional on a Lie algebra transforms when acted upon by elements of the dual group, leading to a geometrical structure in the dual space. This notion is crucial for understanding various properties of symplectic geometry and representations of Lie groups.
Connected Orbit: A connected orbit is a subset of the phase space in which the action of a Lie group on a manifold generates a path that remains continuous and unbroken. In this context, it highlights the idea that elements related by the group action can be connected through continuous transformations, emphasizing the relationship between symmetry and geometric structures within the manifold. Connected orbits are essential for understanding how Lie groups operate on manifolds and the structure of the resulting quotient spaces.
Effective Action: Effective action refers to a way in which a Lie group acts on a manifold such that the action is not only well-defined but also faithfully represents the elements of the group in terms of their influence on the space. This means that the action can distinguish between different elements of the group and is often related to the orbit structure of points in the manifold, helping to understand how these points are transformed under the group's operations.
Fixed Point: A fixed point is a point in a space that remains unchanged under the action of a given transformation, particularly in the context of group actions. In Lie groups, this concept is important because it helps us understand how these groups interact with various spaces and how their actions can stabilize certain points within those spaces. Fixed points are crucial when analyzing orbits and the behavior of systems under symmetry transformations.
Free orbit: A free orbit refers to the action of a Lie group on a manifold where each element of the group acts freely on the manifold, meaning that no element of the manifold is fixed under the group action, except for the identity element. This concept connects to the study of orbits formed by the action of a Lie group on a space, highlighting the relationship between symmetry and geometry.
Lie Group Action: A Lie group action is a smooth and continuous operation of a Lie group on a manifold, which allows for the exploration of the manifold's structure and its symmetries. This action can be thought of as a way for the elements of a Lie group to 'move' points in the manifold while preserving certain geometric properties. Understanding Lie group actions is crucial for analyzing orbits, stabilizers, and the classification of symmetric spaces, as they reveal how groups interact with geometrical objects.
Linear action: A linear action refers to a way in which a Lie group acts on a vector space such that the group elements transform vectors linearly. This means that if you have a vector and apply a group element to it, the result is still a vector in the same space, and this transformation respects both addition and scalar multiplication. Linear actions are essential in understanding how Lie groups interact with vector spaces and play a crucial role in the study of representations and orbits.
Orbit: An orbit is the set of points that can be reached by the action of a Lie group on a point in a manifold. This concept reflects how elements of a Lie group can transform or move points in a given space, forming a path or 'orbit' that captures the symmetry properties of the group. Understanding orbits is essential to grasp how actions of groups can partition spaces into distinct equivalence classes based on their transformations.
Orbit space: Orbit space is a mathematical construct that arises when analyzing the action of a group on a set, particularly in the context of Lie groups. It consists of the collection of orbits formed by the action of the group on the set, where each orbit represents a distinct equivalence class of points that can be transformed into one another via the group action. Understanding orbit space allows for a deeper comprehension of symmetry and the structure of manifolds, especially in the study of geometric properties and classifications.
Orbit-stabilizer theorem: The orbit-stabilizer theorem is a fundamental result in group theory that relates the size of the orbit of an element under a group action to the size of the stabilizer subgroup of that element. Specifically, it states that for a group acting on a set, the size of the orbit of an element is equal to the index of its stabilizer in the group. This theorem helps in understanding how groups act on spaces and can be used to analyze symmetric structures.
Quotient Space: A quotient space is a construction in mathematics that identifies certain points in a topological space and treats them as a single point, effectively partitioning the space into equivalence classes. This concept is key in understanding structures like quotient Lie algebras and the action of Lie groups, as it allows for the simplification of complex algebraic structures by grouping related elements together.
Representation Theory: Representation theory is the study of how algebraic structures, like Lie algebras and Lie groups, can be represented through linear transformations of vector spaces. This concept connects abstract mathematical entities to more concrete linear algebra, enabling the analysis of their properties and behaviors in various contexts, such as geometry and physics.
So(3): The term so(3) refers to the Lie algebra of the special orthogonal group SO(3), which consists of all skew-symmetric 3x3 matrices. This Lie algebra plays a crucial role in various fields, such as describing rotational symmetries in physics and geometry. Understanding so(3) involves its relationship with tangent spaces, the exponential map, and applications in quantum mechanics and general relativity.
Stabilizer: In the context of Lie group actions, the stabilizer of a point is the subset of elements in the Lie group that leave that point unchanged under the group's action. This concept connects the structure of the group with the geometry of the space on which it acts, revealing important properties about orbits and symmetry.
Su(2): su(2) is a Lie algebra that consists of all anti-Hermitian $2 \times 2$ matrices with trace zero. It is crucial in understanding various areas of mathematics and physics, as it provides the algebraic structure underlying the special unitary group SU(2), which is significant in the study of symmetries, particularly in quantum mechanics and particle physics.
Transitive action: Transitive action refers to a type of group action where a group acts on a set in such a way that for any two points in that set, there exists an element of the group that can map one point to the other. This concept highlights the idea of symmetry and interconnectedness within the structure of groups and spaces, illustrating how elements can be transformed into one another through group operations. It serves as a key feature in understanding orbits and the classification of homogeneous spaces, emphasizing the relationship between group elements and their actions on sets.