The Exponential Map

Abstraction of Exponentials

We know exponentials and logarithms as functions . Here, we shall generalize them in an abstract way to the setting of formal power series over a field.

Exponential and Logarithm in

Suppose is a field, consider the ring of formal power series over : Then the exponential and logarithm are elements defined as follows: where is the ideal of generated by .

Proposition

Suppose and are two elements in , and , then is well-defined and belongs to .

Proof Suppose and . Substitute into , we getNote that has lowest degree term of degree , so the th coefficient of is always a finite sum, therefore is well-defined and belongs to .

Corollary

is well-defined and equals for all . is also well-defined and equals for all .

Proof This is a direct consequence of the above proposition.

Exponential and Logarithm of Complex Matrices

We now define the exponential and logarithm of complex matrices by substituting matrices into the formal power series defined above.

Proposition

The following properties hold for the exponential and logarithm maps for complex matrices:

1. For all with , we have .
2. For all with , we have .
3. .
5. if and commute.
6. .

Lie's Product Formula

For any two complex matrices , we have

The Lie Algebra of a Lie Group

Every Lie group has a canonical Lie algebra , which has two natural representations, either as the tangent space at the identity element of , or as the space of left-invariant vector fields on . We will see that the two representations are isomorphic.

Left-Invariant Vector Field

If is a Lie group and is a vector in , where is the identity element in . Suppose is the left multiplication map that sends to . Then we can use the maps to define a natural vector field on , for each . We set

The resulting vector field is called a left-invariant vector field.

e.g.

• Let Lie group , with the group operation of complex number multiplication. Fix some , the left-invariant vector field is given by

Proposition

The left-invariant vector fields on a Lie group form a Lie algebra. That is, suppose and are smooth left-invariant vector fields on , then is also left-invariant.

Proof It suffices to check

Theorem

The set of all left-invariant vector fields on a Lie group is isomorphic to the tangent space at the identity element of , via evaluation at . In other words, every tangent vector extends uniquely to a left-invariant vector field.

Proof Denote the space of left-invariant vector fields as . We define by , and by where . It is clear that and are linear maps, and , . Therefore, and are isomorphisms.

Lie Algebra of a Lie Group

The Lie algebra of a Lie group , often denoted , is up to isomorphic, the Lie algebra of left-invariant vector fields on , or the tangent space at the identity element of .

Remark

Elements of the Lie algebra represent “infinitesimal transformations” or “directions of motion” within the group starting from the identity.

Exponential Map

For any Lie group with Lie algebra , the exponential map is defined by where is the unique integral curve of the left-invariant vector field such that , or equivalently, is the one-parameter subgroup generated by .

e.g. The Lie algebra of , denoted , is isomorphic to . The exponential map of satisfies

Proposition

Let be a Lie group with associated Lie algebra . For any , is the one-parameter subgroup of generated by . That is, for all .

Proof

Fundamental Vector Field

Suppose is a Lie group with Lie algebra , acting on a smooth manifold with the action that . The fundamental vector field on generated by the infinitesimal action of , denoted by (or ), is defined at each point as

e.g. Since the -torus , its Lie algebra is isomorphic to . From the previous example of , we can deduce that the exponential map of an -torus takes the form Consider a torus acting on by for nonzero constants . . Then for any and , there holds