1.1 Definition and examples of Lie groups

Lie groups blend smooth manifolds with group structures, enabling the study of continuous symmetries in mathematics and physics. They provide a powerful framework for analyzing transformations and symmetries in various fields.

Examples of Lie groups include matrix groups like GL(n,ℝ) and SO(n), as well as transformation groups like the Euclidean and Lorentz groups. These illustrate the diverse applications of Lie group theory across different areas of study.

Lie groups and their properties

Definition and essential characteristics

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• A Lie group is a mathematical object that combines the structure of a group with the properties of a smooth manifold
• The group operation (multiplication) and inversion map in a Lie group are required to be smooth (infinitely differentiable) functions between the underlying manifolds
• Lie groups possess a unique identity element, which acts as a neutral element under the group operation
• Each element in a Lie group has a unique inverse with respect to the group operation, allowing for the "undoing" of the operation

Compatibility of group and smooth structures

• The group operation in a Lie group is compatible with its smooth manifold structure, ensuring that the result of the operation remains within the group
• The smoothness property of the group operation and inversion map allows for the application of differential calculus techniques to study the local structure of Lie groups
• The compatibility between the group and smooth structures enables the study of symmetries and continuous transformations in various mathematical and physical contexts

Examples of Lie groups

Matrix Lie groups

• The general linear group $GL(n, \mathbb{R})$ consists of all invertible $n \times n$ real matrices and forms a Lie group under matrix multiplication
• The special linear group $SL(n, \mathbb{R})$ is a subgroup of $GL(n, \mathbb{R})$ consisting of matrices with determinant 1 and is also a Lie group
• The orthogonal group $[O(n)](https://www.fiveableKeyTerm:o(n))$ consists of all $n \times n$ orthogonal matrices (matrices $A$ satisfying $A^T A = I$) and forms a Lie group under matrix multiplication
• The special orthogonal group $SO(n)$ is a subgroup of $O(n)$ consisting of orthogonal matrices with determinant 1 and is also a Lie group
• The unitary group $[U(n)](https://www.fiveableKeyTerm:u(n))$ consists of all $n \times n$ unitary matrices (matrices $U$ satisfying $U^* U = I$, where $U^*$ is the conjugate transpose) and forms a Lie group under matrix multiplication

Transformation groups

• The Euclidean group of isometries of $\mathbb{R}^n$ is a Lie group that describes rigid motions (translations and rotations) in $n$-dimensional Euclidean space
• The Lorentz group is a Lie group of isometries of Minkowski spacetime, which is fundamental in the study of special relativity
• The Poincaré group, a combination of translations and Lorentz transformations, is a Lie group that describes the symmetries of spacetime in special relativity
• The group of diffeomorphisms of a smooth manifold, denoted as $Diff(M)$, is an infinite-dimensional Lie group that plays a crucial role in the study of smooth dynamical systems and fluid dynamics

Smoothness in Lie groups

Differentiability of group operations

• Smoothness in Lie groups refers to the differentiability of the group multiplication and inversion maps
• The group operations in a Lie group are required to be smooth functions between the underlying manifolds, meaning they are infinitely differentiable
• The smoothness property allows for the application of differential calculus techniques, such as the computation of derivatives and the study of tangent spaces, to analyze the local structure of Lie groups

Smooth maps between Lie groups

• A Lie group homomorphism is a smooth map between two Lie groups that preserves both the group structure and the smooth manifold structure
• Lie group homomorphisms are essential in studying the relationships between different Lie groups and their representations
• The composition of Lie group homomorphisms is again a Lie group homomorphism, allowing for the construction of categories of Lie groups and their morphisms
• The study of smooth maps between Lie groups leads to the development of the theory of Lie group actions on smooth manifolds, which has applications in geometry, topology, and mathematical physics

Group operations in Lie group theory

Properties of the group operation

• The group operation in a Lie group, typically denoted as multiplication, combines two elements of the group to produce a third element within the group
• The group operation satisfies the axioms of associativity, meaning that $(ab)c = a(bc)$ for any elements $a$, $b$, and $c$ in the Lie group
• The group operation is compatible with the smooth structure of the Lie group, ensuring that the result of the operation remains within the group and preserves smoothness

Identity element and inverses

• A Lie group possesses a unique identity element, often denoted as $e$, which acts as a neutral element under the group operation, satisfying $ae = ea = a$ for any element $a$ in the group
• Each element in a Lie group has a unique inverse with respect to the group operation, denoted as $a^{-1}$ for an element $a$, satisfying $aa^{-1} = a^{-1}a = e$
• The existence of inverses allows for the "undoing" of the group operation and is crucial in the study of symmetries and transformations described by Lie groups

Interplay between group operation and smooth structure

• The compatibility between the group operation and the smooth manifold structure is a fundamental aspect of Lie group theory
• The group operation and the smooth structure work together to enable the study of continuous symmetries and transformations in various mathematical and physical contexts
• The interplay between the group operation and smooth structure gives rise to the notion of infinitesimal generators of a Lie group, which are elements of the associated Lie algebra and describe the local behavior of the group
• The exponential map, a smooth map from the Lie algebra to the Lie group, connects the infinitesimal generators to the global structure of the group and plays a crucial role in the study of one-parameter subgroups and the representation theory of Lie groups