Sumários

5. Matrix Lie groups and Lie algebras

9 outubro 2020, 11:00 • Pedro Resende

Sections 3.7 and 3.8 of Brian Hall's book — Smooth structure on a matrix Lie group, exponential coordinates. The Lie algebra of a matrix Lie group as the tangent space at the identity. Smoothness of the continuous homomorphisms between matrix Lie groups. Representation of the elements of a connected matrix Lie group by products of exponentials. Corollary: connected matrix lie groups whose Lie algebras are abelian are themselves abelian. Section 3.5 — Functoriality of the Lie algebra of a matrix Lie group: the differential of a homomorphism of matrix Lie groups; the differential of F as the unique Lie algebra homomorphism f that commutes with the exponentials:


exp o f = F o exp.

Properties of the functor Lie : LieGrp -> LieAlg: faithfulness (in the case of connected matrix Lie groups); preservation of kernels. The adjoint representation

Ad : G -> GL(g).

The adjoint representation of Lie algebras ad as the differential of Ad. Section 3.6 — Complexification of real Lie algebras. Section 5.3 — The Baker-Campbell-Hausdorff (BCH) formula. Section 5.7 — Introduction to the problem of integrating homomorphisms of Lie algebras: obtaining local homomorphisms of matrix Lie groups from homomorphisms of their Lie algebras.

4. Lie algebras

8 outubro 2020, 11:00 • Pedro Resende

Exercises from Brian Hall's book: Chapter 2, ex. 6, 7, 8, 9, 10; Chapter 3, ex. 2, 5. Existence of a locally defined homeomorphism between a matrix Lie group (around the identity) and its Lie algebra (around 0): Theorem 3.42 (section 3.7).

3. Lie algebras

2 outubro 2020, 11:00 • Pedro Resende

(Roughly sections 2.1–2.4, 3.1, 3.3–3.4 of Brian Hall's book.) Introduction to Lie algebras. Homomorphisms, subalgebras, ideals, and quotients. Lie algebras of derivations. Representations of Lie algebras. The adjoint representation of a Lie algebra on itself. Faithful representations and Lie algebras of matrices. "Easy" faithful representations: abelian Lie algebras; Lie algebras with trivial center. Statement of Ado's theorem. Matrix exponentials and logarithms. One-parameter subgroups of a matrix Lie group. The Lie algebra of a matrix Lie group. Examples.

2. Lie groups and matrix Lie groups

1 outubro 2020, 11:00 • Pedro Resende

Exercises from chapter 1 of Brian Hall's book: 5, 9, 10, 13, 14, 17, 18.

1. Lie groups and matrix Lie groups

25 setembro 2020, 11:00 • Pedro Resende

Definition of Lie group, homomorphism, isomorphism, Lie subgroup, matrix Lie group. Examples, including the general and special linear groups, orthogonal, special orthogonal, unitary and special unitary groups, generalized orthogonal groups, Lorenz group, symplectic groups, Euclidian groups, Poincaré groups, compact symplectic groups. Topological properties: compactness, connectedness, simple connectedness. The unit component of a Lie group (respectively matrix Lie group). Examples. SU(2) as the 3-sphere, SO(3) as projective 3-space. The double cover SU(2)->SO(3).

This lecture corresponds closely to chapter 1 of Brian Hall's book minus section 1.2.8.
Exercises for next week (from that chapter): 4, 5, 9, 10, 13, 14, 16, 17, 18.