Lie Group

A Lie group is a mathematical group that is also a smooth manifold, with multiplication and inversion given by smooth maps. It combines the algebraic study of symmetry in group theory with the methods of differential geometry. Rotations, translations, and many families of invertible matrices are Lie groups. Their central feature is that finite transformations can be studied through infinitesimal transformations, organized into an associated Lie algebra. (ocw.mit.edu)

Definition and interpretation

A finite-dimensional real Lie group $G$ consists of a group and a finite-dimensional smooth manifold structure such that

and

are smooth. Under the usual conventions, the manifold is Hausdorff and second countable. The group axioms require associativity, an identity element $e$, and an inverse for every element. Its dimension is its manifold dimension: the number of independent local parameters needed to describe an element. (ocw.mit.edu)

Smooth compatibility is stronger than the continuity required of a topological group. It allows differentiation of group operations. Left and right translations,

are diffeomorphisms, so every point has the same local geometric structure as the identity. Lie groups need not be connected or commutative. Countable discrete groups also qualify as zero-dimensional Lie groups under these conventions. A complex Lie group instead has a complex manifold structure with holomorphic group operations. (ocw.mit.edu)

Fundamental examples

Many examples are matrix Lie groups, whose operations are matrix multiplication and inversion. The general linear group $GL(n,\mathbb R)$ consists of invertible real $n\times n$ matrices; it is an open subset of the space of all such matrices and has dimension $n^2$. Important subgroups are defined by preserving additional structures. (ocw.mit.edu)

Here $A^\dagger$ denotes the conjugate transpose. The orthogonal and unitary groups preserve real Euclidean and complex Hermitian inner products, respectively. Although their elements are complex matrices, the dimensions listed for $U(n)$ and $SU(n)$ are real dimensions. (ocw.mit.edu)

The circle group $U(1)=\{z\in\mathbb C:|z|=1\}$ describes planar rotations. The group $SO(3)$ describes spatial rotations, while $SE(3)$, the special Euclidean group, combines rotations and translations into orientation-preserving motions of a rigid body. It has dimension six: three rotational and three translational parameters. (ocw.mit.edu)

Lie algebra and infinitesimal structure

The Lie algebra of $G$, conventionally written $\mathfrak g$, is the tangent space at the identity:

It is a vector space equipped with a bilinear operation called the Lie bracket. A tangent vector at $e$ extends uniquely to a left-invariant vector field, and the commutator of these fields defines the bracket. It satisfies

and the Jacobi identity,

These identities distinguish a Lie algebra from an arbitrary algebra with a bilinear product. (ocw.mit.edu)

For matrix Lie groups, the bracket is

Thus $\mathfrak{gl}(n,\mathbb R)$ contains all real matrices, $\mathfrak{sl}(n,\mathbb R)$ contains traceless matrices, and $\mathfrak{so}(n)$ contains skew-symmetric matrices. The algebra $\mathfrak{su}(n)$ consists of traceless skew-Hermitian matrices. The bracket measures the infinitesimal failure of transformations to commute; in a connected Lie group, a zero bracket is equivalent to commutativity of the group. (ocw.mit.edu)

Exponential map

Each $X\in\mathfrak g$ determines a unique smooth one-parameter subgroup $\gamma_X:\mathbb R\to G$ satisfying

The exponential map is defined by

For matrix groups this is the matrix exponential,

It turns an infinitesimal generator into a finite transformation. Its derivative at zero is the identity, so it is a diffeomorphism between suitable neighborhoods of $0$ and $e$. (ocw.mit.edu)

The exponential is not generally a group homomorphism from vector addition: $\exp(X+Y)=\exp(X)\exp(Y)$ holds when $X$ and $Y$ commute, but not in general. Locally, the Baker–Campbell–Hausdorff formula expresses the correction:

Nor is the exponential necessarily globally injective or surjective. It is surjective for connected compact Lie groups, but connectedness alone does not guarantee surjectivity. Every element of a connected Lie group can nevertheless be expressed as a finite product of exponentials. (ocw.mit.edu)

Local structure and global topology

The fundamental correspondence between groups and algebras has precise limits. Every finite-dimensional real Lie algebra is the Lie algebra of a connected, simply connected Lie group, unique up to isomorphism. Every connected Lie group with that algebra is a quotient of this simply connected group by a discrete central subgroup. Consequently, the algebra determines local multiplication but not all global topological information. (ocw.mit.edu)

For example, $\mathbb R$ under addition and the circle group have the same one-dimensional abelian Lie algebra, but different global topology. Likewise, $SU(2)$ and $SO(3)$ have isomorphic Lie algebras, while

The first is simply connected and double-covers the second. This distinction matters when deciding which algebra representations extend to representations of a particular group. (ocw.mit.edu)

The identity component $G^\circ$ of a Lie group is an open normal subgroup. The Lie algebra captures this connected component locally, but does not by itself specify the other components or how they interact with it. (ocw.mit.edu)

Subgroups, actions, and representations

The closed subgroup theorem states that every closed subgroup of a Lie group is an embedded Lie subgroup. Closedness is important: a subgroup need not be an embedded submanifold. For instance, a one-parameter subgroup can wind densely through a two-dimensional torus. Such phenomena require distinguishing embedded from immersed Lie subgroups. (ocw.mit.edu)

A smooth group action lets a Lie group act by transformations on a manifold. If $H$ is a closed subgroup, the space of cosets $G/H$ is a smooth homogeneous space, with

The sphere, for example, can be expressed as $S^2\cong SO(3)/SO(2)$: rotations act transitively, and rotations about a fixed axis stabilize a point. (ocw.mit.edu)

A finite-dimensional representation is a smooth homomorphism

Differentiation gives a Lie algebra representation $d\rho:\mathfrak g\to\mathfrak{gl}(V)$. Representation theory studies such linear realizations of symmetry. Compact groups have especially strong properties: their finite-dimensional complex representations admit invariant positive-definite Hermitian inner products and decompose into irreducible representations. (ocw.mit.edu)

Structure and classification

Lie groups are often distinguished as compact or noncompact, connected or disconnected, and abelian, solvable, or semisimple. A Lie algebra is semisimple when it has no nonzero solvable ideals; it is simple when it is nonabelian and has no nonzero proper ideals. These conditions make classification substantially more rigid than for general Lie algebras. (math.stanford.edu)

Finite-dimensional complex simple Lie algebras fall into four classical families,

and five exceptional types,

Their structure is encoded by root systems and Dynkin diagrams. For compact connected groups, maximal tori—maximal connected abelian subgroups—provide a central organizing tool; their dimension is the group's rank. Classification of groups requires additional global data beyond the Lie algebra, including central quotients. Noncompact real forms introduce further distinctions. (math.stanford.edu)

Historical development

The subject is named after Sophus Lie (1842–1899), who developed continuous transformation groups in connection with differential equations. During the winter of 1873–1874, he began developing this theory systematically. His “infinitesimal groups” anticipated what are now called Lie algebras: structures describing the infinitesimal generators of transformations rather than groups in the modern sense. The original transformation-based viewpoint remains important in the study of symmetries of differential equations. (mathshistory.st-andrews.ac.uk)

Applications and scope

In robotics, $SO(3)$ and $SE(3)$ describe orientations and configurations without treating them as unconstrained Euclidean vectors. Their Lie algebras describe angular velocities and rigid-body twists; exponential maps relate these velocities to finite displacements. The same framework supplies coordinate transformations and the mathematical basis of robot kinematics. (modernrobotics.northwestern.edu)

In quantum field theory, continuous internal symmetries are organized using Lie groups and their representations. The Standard Model is conventionally described using the gauge-group product $SU(3)\times SU(2)\times U(1)$. Matter fields transform in representations, while infinitesimal gauge transformations use the corresponding Lie algebras. (tpi.uni-jena.de)

The standard finite-dimensional theory does not automatically cover all continuously varying transformation groups. Groups of diffeomorphisms and spaces of gauge transformations are generally infinite-dimensional and require additional analytic structures. Even within finite dimensions, infinitesimal calculations must be supplemented by global information when questions involve covering groups, disconnected components, or which representations exist on a specified group. (ocw.mit.edu)

References

1. 745 F20 Lecture 02: Lie Groups Iocw.mit.edu
2. 745 F20 Lecture 03: Lie Groups IIocw.mit.edu
3. 745 F20 Lecture 04: Homogeneous Spaces and Lie Group Actionsocw.mit.edu
4. 745 F20 Lecture 06: Classical Lie Groupsocw.mit.edu
5. 745 F20 Lecture 07: The Exponential Map of a Lie Groupocw.mit.edu
6. 745 F20 Lecture 08: Lie Algebrasocw.mit.edu
7. 745 F20 Lecture 09: Fundamental Theorems of Lie Theoryocw.mit.edu
8. Lie Algebrasmath.mit.edu
9. Math 210C: Compact Lie Groups — Lecture Notesmath.stanford.edu
10. Math 210C: Lie Theorymath.stanford.edu
11. Sophus Lie (1842–1899) — Biographymathshistory.st-andrews.ac.uk
12. Chapter 3 — Modern Roboticsmodernrobotics.northwestern.edu