**Lie Groups:**

**Definition 1.** A **Lie group** is a manifold and group whose binary operation is a map between the manifolds and

We will assume our Lie groups are Banach manifolds unless otherwise stated (locally homeomorphic to a Banach space ). It turns out a Lie group is metrizable, and the metric space is complete. Lie subgroups are subgroups which are also submanifolds and Lie groups.

**Proposition 2.** Let be an -Lie group (that is, a Lie group which is an -manifold). Then

- If then is locally connected.
- If then
- Suppose is locally compact, then is locally compact iff
- If is generated by a subspace whose topology admits a countable base, then the topology on admits a countable base.

Now suppose is a -manifold with an analytic associative binary operation with identity. Then the subset of invertible elements of is open in and is a Lie group.

**Definition 3.** A **Lie group morphism** between Lie groups and is a group homomorphism will denote the group of Lie group automorphisms of A **Lie group representation** on a Banach space is a Lie group morphism (Note if is a Banach space over the field , then is a Lie group which is a manifold over )

**Definition 4.** Let be a topological group and be an analytic manifold. We say that is a **homogeneous space** for if is a -space (that is, continuous action) and

This induces an equivalence relation on into orbits under the action of and a corresponding quotient space called the **orbit space** of by

**Proposition 5.** Let be Lie group and be a Lie subgroup. Then:

- The orbit space has a unique analytic manifold structure such that the canonical map is a submersion.
- If then is a Lie group and is a Lie group morphism.

In case 1 above, we call the quotient the **Lie homogeneous space** of by and in case 2 we call the quotient the **Lie quotient group**.

Let and be Lie groups. Then there is a canonical correspondence between and In particular, suppose we have two tangent vectors and Let us then define

It follows that for

Thus we have a product on the tangent bundle of (note that being a group makes an arbitrary element of ). Hence is a Lie group.

**The Lie Algebra of a Lie Group:**

Recall a vector field on a manifold is section of the vector fiber bundle with as its base. Hence it is a map satisfying where is the projection of the vector fiber bundle to its base. Now suppose and are two vector fields on then we can define the vector fields

where This turns the set of vector fields of into a Lie algebra.

Now let be a Lie group and be left multiplication by These are certainly diffeomorphisms of

**Definition 6.** A vector field is **left-invariant** if

for all where

**Proposition 7.** The set of left-invariant vector fields of is a Lie subalgebra of

*Proof.* Let and be left-invariant and . Then

and

The Lie subalgebra of is called the **Lie algebra of** Note an obvious consequence is that if and Then

Hence it follows that left-invariant vector fields of are completely determined by where they send the identity element. In this sense, we obtain the equivalence of the vector spaces and

**Lie Group of a Lie Algebra:**

The standard convention is to discuss an “inverse” map from the Lie algebra of a Lie group back to the Lie group. Let Then this left-invariant vector field corresponds to a tangent vector Let be a path defined by

Note these uniquely define called a **one-parameter subgroup** of We in turn define the map by

or more generally

Hence It follows that

Suppose Then and

[1] Bourbaki, Nicholas. *Lie Groups and Lie Algebras, Chapters 1-3*. Springer-Verlag. 1971.

[2] Robert Milson, Thomas Foregger, Mike Fikes. “Lie group” (version 12). *PlanetMath.org*. Freely available at http://planetmath.org/?op=getobj;from=objects;id=1112