# Derivations, vector fields, Lie groups and their Lie algebras

In this post, we will add a bunch of examples to those we saw in last post. We will do a brief excursion into the world of differential geometry, where we will see how the idea of a Lie algebra is related to Lie groups and geometry. If you’ve never seen the definition of a smooth manifold, I refer you to the Wikipedia page.

First, remember that given a algebra a *derivation *of is a linear operator satisfying Leibniz rule: for every

For example, the usual directional derivatives in from calculus are obviously derivations of the algebra of smooth real valued functions We also saw in my first post on Lie algebras that for a Lie algebra and , is a derivation of We will note the set of all derivations of an algebra

It is easy to see that if is an algebra over the field is a vector subspace of In fact, when we equip with the bracket operation (making it a Lie algebra), the subspace is a Lie subalgebra of Indeed, for and we have

Let’s now digress a little bit to describe what is a vector field on a smooth manifold.

Consider a smooth surface sitting in a sphere for example. Then for any parametrized curve such that the derivative is a vector sitting in the plane tangent to at Intuitively, this plane is actually the space of all vectors that are the derivative of some curve with i.e. With this point of view, the tangent space to a point can be viewed as the set of “arrows” emanating from

Alternatively, we can adopt kind of a dual point of view. Define a *derivation at p* to be a a linear map satisfying the Leibniz rule, where is the vector space of *germs* of smooth real valued functions at (i.e. the set of equivalence classes of functions under the relation iff on some open neighborhood of It can be shown that the derivations at are exactly the directional derivatives at given by

for some smooth curve with and Thinking of again as an “arrow” emanating from directing the derivation, we can think of the tangent space of at as the actual set of derivations at This is the easiest way to define the tangent space of an arbitrary smooth manifold.

For a smooth manifold and define the tangent space at noted to be the space of derivations at Equivalently, we can define it as the space of vectors

where is a coordinate map defined on a chart around p and a curve such that

For some manifold the (disjoint) union of all tangent spaces is called the tangent bundle,

There is a natural projection map defined by for a point of and a vector in The space can be given a canonical smooth structure making it itselft a differentiable manifold. A *smooth **vector field* on a differentiable manifold is then a smooth section of i.e. a smooth map such that

The important result that connects this all with what we’ve done so far is that the space of smooth vector fields on is exactly the space of derivations of the real algebra of smooth real-valued functions Therefore, what we have said at the beginning of this post applies :

the set of smooth vector fields on a differentiable manifold form a Lie algebra.

We can now describe Lie groups and their Lie algebras. A Lie group is a group that also has the structure of a smooth manifold in which the operations of multiplication and taking inverses are diffeomorphisms. So on the one hand we can use what we know from group theory on them, but on the other the notions of vector fields and all the machinery from differential geometry can be used to study them.

We can show that left multiplication in a Lie group is a diffeomorphism. We thus have the notion of a left invariant vector field on a Lie group : noting left multiplication by by a vector field is *left invariant* if for all and It can be shown that left invariant vector fields are smooth and closed under the bracket operation and so form a Lie subalgebra of the space of smooth vector fields. This is the Lie algebra of the Lie group

A left invariant vector field is determined locally. For example, if you know how it acts on functions at the identity then you know that for any and it acts on at by

This observation in fact gives us a vector space isomorphism between the space of left invariant vector fields on and the tangent space at the identity : for a vector field take its component in and from a tangent vector at the identity create a left invariant field like above. We can thus transport the Lie algebra structure of left invariant vector fields to the tangent space at the identity and this gives us a more concrete way to view the Lie algebra of a Lie group.

Indeed, consider all smooth curves with Then the Lie algebra of being identified with can be described as with the Lie algebra structure induced by the isomorphism above. When is one of the classical Lie groups of matrices, we recover in this way the classical linear Lie algebras described in my last post. Here are some examples:

The set of invertible matrices can be given a differentiable structure in a natural way : we just embedd it in and use the induced differentiable structure. Then is an open subset, being the preimage of Moreover, multiplication and inversion are given by polynomial functions so they are differentiable operations and forms a Lie group. (Note that this discussion is also valid for instead of A closed subgroup of some is called a *closed linear group*. Examples of closed linear groups are

These are all groups of *continuous *symmetries. For example, is the group of rotations in 3 dimensional Euclidean space. Thus a curve in represents a rotational motion and the set for all curves with ie. the Lie algebra of contains all the information about *infinitesimal* rotational motion. For example, this defines a smooth curve in :taking the derivative at 0 gives

which is indeed a skew-symmetric matrix (i.e. a matrix whose transposed is its own additional inverse). Using other curves, we can show that the Lie algebra of skew-symmetric matrices is contained in the tangent space at the identity of In fact, since for a curve in we have we find that

So the Lie algbera of is the Lie algebra of skew-symmetric real matrices. The bracket operation induced by the left invariant vector fields on is just the usual bracket for linear Lie algebras.

Similar considerations shows that the Lie algebra of is the space of matrices such that the Lie algebra of is the space of matrices with trace etc…

This very intriguing stuff. On the one hand, the study of Lie algebras is very algebraic and on the other, Lie groups are very geometric. There is a nice dance between these two notions and I would like to investigate the geometric side of things a little bit more but the task I have given myself is to understand the classification of (semi-)simple Lie algebras. In the next posts, I will thus resume the algebraic investigation but hopefully I will come back to this when I’m done.

**Reference:**

- Knapp –
*Lie Groups Beyond an Introduction* - Humphreys –
*Introduction to Lie Algebras and Representation Theory* - L. W. Tu –
*An Introduction to Manifolds* - Lee –
*Introduction to Smooth Manifolds.*