1.4 The exponential map and its properties

The exponential map connects Lie algebras to Lie groups, transforming elements from one to the other. It's a smooth function that preserves structure and is locally invertible near the identity, helping us understand how these mathematical objects relate.

This map is key for studying Lie groups through their algebras. It's used to compute group elements, find one-parameter subgroups, and explore local structure. Understanding its properties is crucial for grasping the interplay between Lie groups and algebras.

Exponential Map: Lie Groups and Algebras

Definition and Connection

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• The exponential map is a smooth map from the Lie algebra $\mathfrak{g}$ of a Lie group $G$ to the Lie group itself, denoted as $\exp: \mathfrak{g} \rightarrow G$
• For a matrix Lie group, the exponential map is defined using the matrix exponential: $\exp(X) = \sum_{n=0}^{\infty} \frac{X^n}{n!}$, where $X$ is an element of the Lie algebra
• The exponential map transforms elements from the Lie algebra (tangent space at the identity) to the Lie group, establishing a connection between the two structures
• The exponential map is a local diffeomorphism near the identity element of the Lie group, meaning it is smooth and has a smooth inverse in a neighborhood of the identity ($\mathfrak{sl}(2, \mathbb{R})$ and $\mathrm{SL}(2, \mathbb{R})$)

Local Diffeomorphism and Surjectivity

• The exponential map is a local diffeomorphism in a neighborhood of the identity element
  • There exists an open set $U$ containing $0$ in the Lie algebra $\mathfrak{g}$ such that $\exp$ restricted to $U$ is a diffeomorphism onto an open set $V$ containing the identity in the Lie group $G$
• For a connected Lie group $G$, the exponential map is surjective
  • Every element of $G$ can be expressed as the exponential of some element in the Lie algebra $\mathfrak{g}$
• The exponential map is not necessarily injective globally, as there may be multiple elements in the Lie algebra that map to the same element in the Lie group ($\mathfrak{so}(3)$ and $\mathrm{SO}(3)$)
• The exponential map preserves the structure of the Lie algebra, such as the Lie bracket: $\exp([X, Y]) = \exp(X) \exp(Y) \exp(-X) \exp(-Y)$ for $X, Y$ sufficiently close to $0$ in the Lie algebra

Properties of the Exponential Map

Structure Preservation

• The exponential map preserves the structure of the Lie algebra, such as the Lie bracket: $\exp([X, Y]) = \exp(X) \exp(Y) \exp(-X) \exp(-Y)$ for $X, Y$ sufficiently close to $0$ in the Lie algebra
• The exponential map is a homomorphism from the Lie algebra to the Lie group when restricted to a sufficiently small neighborhood of $0$ in the Lie algebra
• The exponential map relates the Lie bracket in the Lie algebra to the commutator in the Lie group: $[\exp(X), \exp(Y)] = \exp([X, Y])$ for $X, Y$ sufficiently close to $0$ in the Lie algebra

Injectivity and Surjectivity

• The exponential map is not necessarily injective globally, as there may be multiple elements in the Lie algebra that map to the same element in the Lie group ($\mathfrak{so}(3)$ and $\mathrm{SO}(3)$)
• For a connected Lie group $G$, the exponential map is surjective
  • Every element of $G$ can be expressed as the exponential of some element in the Lie algebra $\mathfrak{g}$
• The exponential map is a local diffeomorphism near the identity element of the Lie group, meaning it is smooth and has a smooth inverse in a neighborhood of the identity ($\mathfrak{sl}(2, \mathbb{R})$ and $\mathrm{SL}(2, \mathbb{R})$)

Applying the Exponential Map

Computing Exponentials

• Use the exponential map to compute the exponential of specific elements in the Lie algebra and find their corresponding elements in the Lie group
• For matrix Lie groups, the matrix exponential can be calculated using the power series: $\exp(X) = \sum_{n=0}^{\infty} \frac{X^n}{n!}$ ($\mathfrak{gl}(n, \mathbb{R})$ and $\mathrm{GL}(n, \mathbb{R})$)
• Utilize properties of the exponential map, such as linearity and the exponential of a commutator, to simplify calculations

One-Parameter Subgroups

• Utilize the exponential map to determine one-parameter subgroups of a Lie group by exponentiating elements of the Lie algebra
• A one-parameter subgroup of a Lie group $G$ is a smooth homomorphism $\phi: \mathbb{R} \rightarrow G$, where $\mathbb{R}$ is the additive group of real numbers
• For every element $X$ in the Lie algebra $\mathfrak{g}$, the exponential map defines a one-parameter subgroup $\phi_X: \mathbb{R} \rightarrow G$ given by $\phi_X(t) = \exp(tX)$ for all $t \in \mathbb{R}$

Local Structure and Baker-Campbell-Hausdorff Formula

• Employ the exponential map to study the local structure of a Lie group near the identity by examining the behavior of the exponential map in a neighborhood of $0$ in the Lie algebra
• Apply the Baker-Campbell-Hausdorff formula, which relates the exponential of a sum of Lie algebra elements to the product of their exponentials, to simplify calculations involving the exponential map: $\exp(X) \exp(Y) = \exp(X + Y + \frac{1}{2}[X, Y] + \cdots)$ for $X, Y$ sufficiently close to $0$ in the Lie algebra
• Use the Baker-Campbell-Hausdorff formula to approximate group operations in terms of Lie algebra operations

Exponential Map and One-Parameter Subgroups

One-Parameter Subgroups

• A one-parameter subgroup of a Lie group $G$ is a smooth homomorphism $\phi: \mathbb{R} \rightarrow G$, where $\mathbb{R}$ is the additive group of real numbers
• For every element $X$ in the Lie algebra $\mathfrak{g}$, the exponential map defines a one-parameter subgroup $\phi_X: \mathbb{R} \rightarrow G$ given by $\phi_X(t) = \exp(tX)$ for all $t \in \mathbb{R}$
• Conversely, every one-parameter subgroup of a Lie group $G$ is of the form $\exp(tX)$ for some element $X$ in the Lie algebra $\mathfrak{g}$

Correspondence between Lie Algebra and One-Parameter Subgroups

• The exponential map establishes a bijective correspondence between the elements of the Lie algebra and the one-parameter subgroups of the Lie group
• This correspondence allows for the study of the infinitesimal behavior of a Lie group through its one-parameter subgroups, which are determined by the elements of the Lie algebra
• The exponential map relates the Lie algebra, which is a linear space, to the one-parameter subgroups, which are curves in the Lie group
• The correspondence between Lie algebra elements and one-parameter subgroups facilitates the analysis of the local structure of a Lie group