Playlist - Lie groups (Borcherds)

We give an introductory survey of Lie groups theory by describing some examples of Lie groups in low dimensions.

We define the Lie algebra of a Lie group in two ways, and show that it satisfies the Jacobi identity. The we calculate the Lie algebras of a few Lie groups.

We discuss the relation between Lie groups and Lie algebras, and give several examples showing how they behave differently. Lie algebras turn out to correspond more closely to the simply connected Lie groups. We then explain why any connected Lie groups is the quotient of a simply connected Lie group (its universal covering space) by a discrete subgroup, and work out some examples of universal covering groups.

We define the exponential map for matrix groups and describe its basic properties. (We also sketch two ways to define it for general Lie groups.) We give an example to show that it need not be surjective even for connected groups, and an example to show that for infinite dimensional groups there can be problems with its convergence.

We state the Baker-Campbell-Hausdorff formula for exp(A)exp(B). As applications we show that a Lie group is determined up to local isomorphism by its Lie algebra, and homomorphisms from a simply connected Lie group are determined by homomorphisms from its Lie algebra. We show how to prove the BCH formula using primitive elements of the Hopf algebra of non-commutative power series. Finally we display an explicit form of the BCH formula.

We state the Poincaré-Birkhoff Witt theorem, which shows that the universal enveloping algebra (UEA) of a Lie algebra is the same size as a polynomial algebra. We prove it for Lie algebras of Lie groups and sketch a proof of the general case.

As an application we show that in characteristic 0 the primitive elements of the UEA are just the original Lie algebra. The case when L is a free Lie algebra on two generators was used in the proof of the Baker-Campbell-Hausdorff formula in the previous lecture.

We give several examples to show that, over fields of positive characteristic, Lie algebras can behave strangely, and have a weaker connection to Lie groups. In particular the Lie algebra does not generate the ring of all invariant differential operators.

We give a sketch of the Bianchi classification of the Lie algebras and groups of dimension at most 3. We mention that this is related to the Thurston geometries of 3-manifolds.

We state Engel's theorem about nilpotent Lie algebras and sketch a proof of it. We give an example of a nilpotent Lie group that is not a matrix group.

This lecture is about Lie's theorem, which implies that a complex solvable Lie algebra is isomorphic to a subalgebra of the upper triangular matrices.

We show the existence of a left-invariant measure (Haar measure) on a Lie group. and work out several explicit examples of it.