Let be a Lie group. Let be a smooth curve in passing
through the unit element **1**of , i.e., a smooth mapping from a
neighborhood of 0 on the real line into with . , the
tangent space of at **1**, consists of all matrices of the form
, or just in a less clumsy
notation.

is the Lie algebra of . I will show in a moment that is a
vector space over **R**, and I really should (but I won't) define a
binary operation (the Lie bracket) on
.

Proof that is a vector space over **R**: if is a smooth
curve and , then set , . This is also
a smooth curve and . So is closed under
multiplication by elements of **R**. (Note this argument fails for
complex .) Similarly, differentiating (with
, as usual) shows that is closed under addition.

Historically, the Lie algebra arose from considering elements of
``infinitesimally close to the identity''. Suppose
is
very small, or (pardon the expression), ``infinitesimally small''. Then
is approximately
, or
(remembering )

Historically, is a so-called infinitesimal generator of .

Robinson has shown how this classical approach can be made
rigorous, using non-standard analysis. Even without this, the classical
notions provide a lot of insight. For example, let be an ``infinite''
integer. Then if is an ordinary real number (not
``infinitesimal''), we can let and so

Assume that the left hand side is an ordinary ``finite'' element of . Write for , an arbitrary element of the Lie algebra . This suggests there should be a map from into .

In fact, the following is true: for any Lie group with Lie algebra , we have a mapping from into such that , and , for any and .

It also turns out that the Lie algebra structure determines the Lie group structure ``locally'': if the Lie algebras of two Lie groups are isomorphic, then the Lie groups are locally isomorphic. Here, the Lie algebra structure includes the bracket operation, and of course one has to define local isomorphism.

Now for our standard example, . Notation: the Lie algebra of is . If you differentiate the condition and plug in , you will conclude that all elements of are anti-symmetric. Fact: the converse is true.

Example: , rotations in 2-space. All elements of have the
form

An element of can be thought of as an angular speed.

In the earlier discussion of , I set up a one-one correspondence

between matrices and complex numbers. (We had the restriction , but we can drop this and still get an isomorphism between matrices of this form and

, rotations in 3-space. Elements of have the form:

(We'll see the reason for the peculiar choice of signs and arrangement of , , and shortly.)

Fact: the vector is the angular velocity vector for the above element of . What does this mean? Well first, let be some arbitrary vector; if is a curve in , and we set , then is a rotating vector, whose tip traces out the trajectory of a moving point. The velocity of this point at is . It turns out that equals the cross-product , which characterizes the angular velocity vector. The next few paragraphs demonstrate this equality less tediously than by direct calculation.

Let

so the general element of can be written . And , , and are simply the elements of corresponding to unit speed uniform rotations about the x, y, and z axes, respectively-- as can be seen by considering their effects on the standard orthonormal basis.

This verifies the equation for the special cases of , and . The general case now follows by linearity.

© 2001 Michael Weiss