Hamiltonian group action

Definition

A Hamiltonian group action is a smooth action of a Lie group on a symplectic manifold that preserves the symplectic structure and is generated by a Hamiltonian function. This concept connects the dynamics of the system with geometric properties, allowing for the analysis of symplectic manifolds in the context of group actions. It plays a crucial role in understanding how symmetries influence the geometry of complex algebraic varieties and in defining symplectic quotients.

5 Must Know Facts For Your Next Test

1. Hamiltonian group actions preserve the symplectic structure, meaning they keep the area form invariant under the group's action.
2. The moment map is central to Hamiltonian group actions as it encodes information about the symmetries and can be used to compute symplectic quotients.
3. Hamiltonian group actions lead to reduced spaces where one can study the dynamics on a lower-dimensional manifold, often simplifying complex systems.
4. In the context of complex algebraic varieties, Hamiltonian group actions help understand how these varieties behave under symmetries, providing insights into their geometric properties.
5. Symplectic quotients derived from Hamiltonian group actions give rise to spaces that can have rich geometrical and topological structures, often facilitating further analysis.

Review Questions

• How does a Hamiltonian group action relate to the preservation of symplectic structures on manifolds?
  • A Hamiltonian group action is defined such that it preserves the symplectic structure of a manifold during its action. This means that as the Lie group acts on the manifold, the associated 2-form remains invariant. This preservation is crucial for studying Hamiltonian dynamics, as it ensures that key geometric properties are retained even as one examines different trajectories or orbits produced by the group's action.
• Discuss the significance of moment maps in the context of Hamiltonian group actions and their relation to symplectic quotients.
  • Moment maps serve as a bridge between Hamiltonian group actions and symplectic quotients. They capture how much the group's action influences the symplectic structure and provide critical information about fixed points and orbits. By taking a moment map into account, one can define symplectic quotients, which result in reduced spaces that inherit a natural symplectic structure from the original manifold, thus facilitating further study of these quotient spaces.
• Evaluate how Hamiltonian group actions impact our understanding of complex algebraic varieties through their geometric and dynamical properties.
  • Hamiltonian group actions significantly enhance our understanding of complex algebraic varieties by linking algebraic structures with geometric dynamics. The preservation of symplectic forms under these actions provides insights into how varieties respond to symmetries, affecting their geometric properties and leading to new invariants. Furthermore, studying these actions helps illuminate connections between algebraic geometry and dynamical systems, allowing mathematicians to apply techniques from both fields to tackle problems related to stability, deformation, and classification of varieties.

Related terms

Symplectic Manifold:

A smooth manifold equipped with a closed, non-degenerate 2-form, which allows for the study of Hamiltonian dynamics and geometric properties.

Moment Map:

A mathematical tool that associates to each point in a symplectic manifold a value in the dual of the Lie algebra of the group, capturing the symmetry properties under a Hamiltonian group action.

GIT Quotient: Geometric Invariant Theory quotient, which provides a way to construct new geometric spaces from old ones by identifying points that are equivalent under group actions.