Primer año de posgradoGeometry & TopologyLección disponible
Lie groups and Lie algebras
Groups that are also smooth manifolds: Lie groups and their Lie algebras, the exponential map, the correspondence between Lie subgroups and subalgebras, the adjoint representation, compact groups and Haar measure, representations of sl(2, ℂ) and a first look at root systems.
math.geom.lie-groups
Se basa en
Requisitos previos obligatorios, hasta el fondo: 45
Aprenderás a
- Define Lie groups and Lie algebras and compute the Lie algebras of classical groups.
- Use the exponential map and the Lie correspondence between connected Lie subgroups and Lie subalgebras.
- Classify the finite-dimensional irreducible representations of sl(2, ℂ) and describe the root system of sl(3, ℂ).
Definición
- A Lie group is a group that is a smooth manifold such that multiplication and inversion are smooth.
- A Lie algebra is a vector space with a bilinear, antisymmetric bracket satisfying the Jacobi identity.
- The adjoint representation of G is the action of G on its Lie algebra by differentiating conjugation.
Teorema
- The tangent space at the identity of a Lie group is a Lie algebra, and Lie group homomorphisms induce Lie algebra homomorphisms.
- Every finite-dimensional real Lie algebra is the Lie algebra of a unique simply connected Lie group (Lie's third theorem).
- For each n ≥ 0 there is exactly one irreducible representation of sl(2, ℂ) of dimension n + 1, up to isomorphism.
- Every compact Lie group has a bi-invariant Haar measure, so its finite-dimensional representations are completely reducible.
Errores frecuentes
- Thinking the exponential map is always surjective; for SL(2, ℝ) the matrix with eigenvalues −2 and −1/2 is not an exponential.
- Believing a Lie algebra determines its Lie group; SU(2) and SO(3) have isomorphic Lie algebras but are not isomorphic.
Conduce a
Descubrimientos opcionales
- Representation theory of finite groups The representation theory of compact Lie groups extends that of finite groups, with integration against Haar measure replacing averaging over the group.
- The Langlands programme: an orientation Langlands duality pairs each reductive group with a dual group whose root system is obtained by exchanging roots and coroots.
- ← The Langlands programme: an orientation
Itinerarios de aprendizaje
Lecturas complementarias
- Brian C. Hall, Lie Groups, Lie Algebras, and Representations (2003)
- James E. Humphreys, Introduction to Lie Algebras and Representation Theory (1972)