Exponential map from Lie algebra to Lie group

Definition

Definition for a real Lie group

Suppose $G$ is a real Lie group and $\mathfrak{g}$ is its Lie algebra. The exponential map from $\mathfrak{g}$ to $G$, i.e., the map:

$\exp: \mathfrak{g} \to G$

is defined as follows. $\exp(x)$ is defined in two steps:

1. First, let $\gamma:\R \to G$ be the unique one-parameter group for which $\gamma'(0) = x$. Note that this makes sense because $\gamma'(0)$ is an element in the tangent space to $G$ at $\gamma(0)$, which is the identity element, and this tangent space is precisely the Lie algebra. The uniqueness of $\gamma$ is less obvious, and requires some use and existence and uniqueness of solutions to ordinary diffferential equations. In fact, $\gamma$ can be defined as a geodesic with respect to any left-invariant connection with tangent vector $x$.
2. We define $\exp(x)$ to be the value $\gamma(1)$.

Definition for a linear Lie group

Suppose $G$ is a linear Lie group over a topological field $K$, i.e., a Lie group with an embedding as a closed subgroup of the general linear group $GL(n,K)$. Its Lie algebra $\mathfrak{g}$ can be naturally identified as a subgroup of the matrix Lie algebra $M(n,K)$.

The exponential map $\operatorname{exp}: \mathfrak{g} \to G$to it is obtained by restricting the matrix exponential $M(n,K) \to GL(n,K)$ to the subalgebra $\mathfrak{g}$.

For a linear Lie group over the reals, this definition coincides with the preceding definition.

Definition for a unipotent algebraic group

Further information: Lie correspondence between nilpotent Lie algebras and unipotent algebraic groups

For a unipotent algebraic group over any field, we can construct its Lie algebra algebraically by taking the Lie algebra of an algebraic group. Note that unipotent algebraic groups are by definition linear, so we can think of the algebraic group as a linear algebraic group and its Lie algebra as a Lie algebra of matrices.

If the characteristic of the field is either zero or is at least as large as the maximum of the nilpotency orders of all elements, we can define an exponential map from the Lie algebra to the group. The map is simply defined by taking the matrix exponential, and noting that by the condition on the characteristic, the series for exponential terminates in finitely many steps.

Unlike the other cases, the exponential map in this case is always a bijection. In fact, this is an algebraic group version of the Lazard correspondence.

Facts

Exponential maps and algebraic groups

For non-unipotent algebraic groups over fields such as the reals, we can construct the Lie algebras algebraically (see Lie algebra of an algebraic group) without resorting to the analytic or topological structure. The Lie algebra thus constructed can be identifid with the Lie algebra of the real Lie group (see Lie algebra of real algebraic group equals Lie algebra of corresponding real Lie group). However, the exponential map from the Lie algebra to the Lie group cannot be defined in a purely algebraic sense. In fact, (is this true?) the exponential map need not in general even be a morphism of algebraic varieties. The reason is that defining the exponential requires some kind of limit-taking or topological procedure that does not make sense in the purely algebraic domain.

On the other hand, for unipotent algebraic groups, the algebraically defined exponential map coincides with the exponential map defined using the matrix exponential. It is a morphism of algebraic varieties.