Cohomology and Extensions
geometric representation theory
representation, 2-representation, ∞-representation
group algebra, algebraic group, Lie algebra
vector space, n-vector space
affine space, symplectic vector space
module, equivariant object
bimodule, Morita equivalence
induced representation, Frobenius reciprocity
Hilbert space, Banach space, Fourier transform, functional analysis
orbit, coadjoint orbit, Killing form
geometric quantization, coherent state
module algebra, comodule algebra, Hopf action, measuring
Geometric representation theory
D-module, perverse sheaf,
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
geometric function theory, groupoidification
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
∞-Lie theory (higher geometry)
Formal Lie groupoids
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).
As symplectic leafs of the Lie-Poisson structure
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.
- B. C., The Structure of the Space of Coadjoint Orbits of an Exponential Solvable Lie Group, ransactions of the American Mathematical Society Vol. 332, No. 1 (Jul., 1992), pp. 241-269, (JSTOR)
Revised on October 29, 2013 23:46:24
by Urs Schreiber