geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Given a Lie group , it acts smoothly on the dual of its Lie algebra by the coadjoint action. The orbits of that action are called coadjoint orbits.
Coadjoint orbits are especially important in the orbit method of representation theory or, more generally, geometric quantization.
Sometimes coadjoint orbits are studied in the infinite-dimensional case (for example in study of Virasoro algebra).
The dual of a (say finite-dimensional real) Lie algebra has a structure of a Poisson manifold with the Poisson structure due to A. Kirillov and Souriau, called the Lie-Poisson structure, namely for any ,
The coadjoint orbits are the symplectic leaves of that structure; hence each orbit is a symplectic manifold.
Last revised on February 25, 2019 at 18:11:48. See the history of this page for a list of all contributions to it.