Lie group

In mathematics, a Lie group (pronounced /liː/ Lee) is a special kind of group that is also a differentiable manifold. This means it has properties of both a group and a smooth, flexible space. Groups help us understand operations like addition or multiplication, and manifolds are spaces that look like regular Euclidean space up close. When these two ideas come together, we get a continuous group where operations can be done smoothly.

Lie groups are important because they help us understand continuous symmetry, like the way a circle can be rotated smoothly. For example, the circle group is a classic Lie group because rotating a circle is a smooth, continuous action. Lie groups are used in many areas of modern mathematics and physics.

These groups were first discovered while studying special kinds of matrix groups called the classical groups. They are named after the Norwegian mathematician Sophus Lie, who developed the theory in the late 1800s. Lie groups were originally created to study the symmetries in differential equations, similar to how finite groups are used in Galois theory to study symmetries in algebraic equations.

History

Sophus Lie thought the winter of 1873–1874 was when his theory of continuous groups began. Others believe his hard work from 1869 to 1873 led to the theory. Lie worked closely with Felix Klein during these years. By 1884, Lie said he had all the main ideas, but his papers were mostly in Norwegian journals, so people in Europe did not notice his work right away.

In 1884, a young mathematician named Friedrich Engel helped Lie write a big book about his theory. The term groupes de Lie was first used in 1893. Lie’s work connected to other areas of mathematics, like solving equations and geometry. Later mathematicians, such as Wilhelm Killing and Élie Cartan, helped organize and expand Lie’s ideas. David Hilbert also asked an important question about Lie’s work in 1900. Hermann Weyl later explained Lie’s theories more clearly and linked them to quantum mechanics.

Overview

Lie groups are special kinds of mathematical shapes that are also groups. This means you can use rules from calculus to study them. They help us understand geometry by showing how certain shapes stay the same when moved or changed in specific ways.

In physics, Lie groups describe symmetries in the natural world. They are important for understanding particles and how they move. These groups also help mathematicians study shapes and their properties in many different ways.

Definitions and examples

A real Lie group is a group that is also a special kind of space called a smooth manifold. This means that the group’s ways of combining elements (like multiplication) and finding opposites (like inverses) are smooth and continuous.

First examples

The 2×2 real invertible matrices form a group under multiplication, called the general linear group of degree 2. This is a four-dimensional real Lie group.
The rotation matrices form a subgroup of the above group, denoted by SO(2). It is a one-dimensional compact connected Lie group.
The affine group of one dimension is a two-dimensional matrix Lie group, consisting of 2 × 2 real, upper-triangular matrices.

Non-example

We now present an example of a group with many elements that is not a Lie group under a certain setup. The group given by a special setup is a subgroup of a torus but is not a Lie group when given a particular type of structure. However, with a different setup, it can be turned into a Lie group.

Matrix Lie groups

Let GL(n, C) denote the group of n × n invertible matrices with entries in C. Any closed subgroup of GL(n, C) is a Lie group; Lie groups of this sort are called matrix Lie groups. The following are standard examples of matrix Lie groups.

The special linear groups over R and C, SL(n, R) and SL(n, C), consisting of n × n matrices with determinant 1 and entries in R or C
The unitary groups and special unitary groups, U(n, C) and SU(n, C), consisting of n × n complex matrices satisfying U* = U−1 (and also det(U) = 1 in the case of SU(n)), where U* is the conjugate transpose of U
The orthogonal groups and special orthogonal groups, O(n, R) and SO(n, R), consisting of n × n real matrices satisfying RT = R−1 (and also det(R) = 1 in the case of SO(n, R)), where RT is the transpose of R

All of the preceding examples fall under the heading of the classical groups.

Related concepts

A complex Lie group is defined in the same way using complex manifolds rather than real ones. Similarly, one can define a p-adic Lie group over the p-adic numbers.

Hilbert's fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples. The answer turned out to be negative. If G is a topological manifold with continuous group operations, then there exists exactly one analytic structure on G which turns it into a Lie group.

Topological definition

A Lie group can also be defined as a topological group that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds. Precisely, a Lie group is defined as a topological group that (1) is locally isomorphic near the identities to a matrix Lie group and (2) has at most countably many connected components.

More examples of Lie groups

Lie groups are found everywhere in math and science. Matrix groups or algebraic groups are groups made of matrices, like orthogonal and symplectic groups. These are some of the most common examples of Lie groups.

Dimensions one and two

The simplest Lie groups with one measurement are the real number line (using addition) and the circle group (using multiplication). The circle group is also called U(1).

For two measurements, there are two main types of Lie groups. One is like moving on a flat plane, and the other is a bit more complex.

Additional examples

The group SU(2) is a group of special 2×2 matrices. It can also be thought of as points on a 3‑sphere.
The Heisenberg group is important in quantum mechanics.
The Lorentz group deals with spaces that mix space and time.
The Poincaré group also works with space and time.
There are special groups called exceptional Lie groups, like G₂, F₄, E₆, E₇, and E₈.
The symplectic group Sp(2n, R) deals with special kinds of 2n×2n matrices.

Constructions

We can make new Lie groups from existing ones in a few standard ways:

Multiplying two Lie groups together gives a new Lie group.
Certain smaller groups inside a Lie group are also Lie groups.
Dividing a Lie group by a special smaller group gives a new Lie group.
The "universal cover" of a Lie group is also a Lie group. For example, the real number line is the universal cover of the circle group.

Related notions

Some groups are not Lie groups because they have too many measurements or because they don’t behave like regular space. These include some groups used in number theory.

Basic concepts

Lie groups are special types of mathematical structures that combine properties of groups and smooth geometries. A group is a set with a rule for combining elements, like adding or multiplying numbers. A smooth geometry, or manifold, is a space that looks flat and simple when you zoom in — like the surface of a sphere or a cube.

When these ideas come together, we get a Lie group: a group that is also a smooth manifold, where the group operations — multiplying elements and finding their opposites — happen in a smooth way. This makes Lie groups very useful for studying continuous symmetries, like rotations and translations.

One key idea connected to Lie groups is the Lie algebra, which captures their local structure near the identity element. Think of it as a simpler version of the group that helps us understand how the group behaves up close.

Representations

Main article: Representation of a Lie group

When studying Lie groups, one important idea is how they can act on spaces filled with arrows, called vector spaces. In physics, Lie groups often show us the symmetries of a system. We can use these symmetries to make problems easier to solve. For example, in quantum mechanics, if a system has a certain symmetry, like the ability to spin around, this can help us understand the system better. This use of symmetry is called representation theory.

When we look at certain Lie groups, we can break down their actions into simpler parts. These simpler parts help us understand the group better. This way of breaking things down was worked out by a mathematician named Hermann Weyl. We can also study how these groups act in more complex ways, but these ideas help make difficult problems much easier.

Classification

Lie groups can be seen as smoothly changing families of symmetries, like rotating around an axis little by little. These tiny changes are studied using a tool called a Lie algebra, which works because Lie groups are smooth shapes with tangent spaces at each point.

Lie groups are grouped based on their properties, such as being simple, connected, or compact. One important result is the Levi decomposition, which shows that a certain type of Lie group can be built from two simpler parts. There are special families of Lie algebras that describe many symmetries, and a few rare ones that don’t fit into these families.

Infinite-dimensional Lie groups

Lie groups are usually studied when they have a limited number of dimensions, but there are also groups that look like Lie groups but have an endless number of dimensions. One way to describe these groups is by using special spaces called Banach spaces, which are similar to the usual Euclidean space but work in more complex ways. However, for many uses, we need to use even more general spaces called locally convex topological vector spaces.

In these more complex spaces, the connection between the Lie algebra (a tool used to study the group) and the Lie group itself becomes trickier, and some properties that work for groups with limited dimensions no longer apply. Some examples of these infinite-dimensional Lie groups include groups of smooth changes on shapes, groups used in theories about space and time, and groups important in advanced physics theories. These groups can have simpler topological properties than their finite-dimensional counterparts.