Lie algebra

A Lie algebra is a special kind of mathematical structure. It helps us study symmetry, which is about how things can twist and turn in relation to each other.

Lie algebras are linked to Lie groups. These are groups that also act like smooth, round shapes. Each Lie group has a Lie algebra that shows its structure up close. This connection makes Lie algebras useful for understanding symmetry in mathematics and physics.

In physics, Lie algebras describe symmetries of space, time, and particles. For example, the ways we can turn three-dimensional space form a Lie group. Their Lie algebra can be shown using the cross product of vectors. This helps us see how small turns can combine. Lie algebras also appear in quantum mechanics, where they explain how particles and forces act.

One common example of a Lie algebra uses square matrices. The Lie bracket is the difference between multiplying two matrices in opposite orders. This shows important ideas about how linear transformations relate to each other.

History

Lie algebras were introduced in the 1870s by Sophus Lie to study small changes in shapes and movements. Later, Wilhelm Killing found them again in the 1880s. The name "Lie algebra" was given by Hermann Weyl in the 1930s. Older books sometimes called them "infinitesimal groups."

Definition of a Lie algebra

A Lie algebra is a special type of mathematical structure called a vector space. It has a special operation called the "Lie bracket" that combines two elements of the space.

This bracket must follow three important rules:

• Bilinearity: This means it behaves nicely with scaling and addition.
• Alternating property: This means combining an element with itself gives zero.
• Jacobi identity: This is a specific relationship that must hold when combining three elements in a certain way.

Lie algebras are often studied with Lie groups, which are continuous groups used in geometry and physics. When a Lie algebra is linked to a Lie group, it is usually written with a special lowercase, stylized letter. For example, the Lie algebra connected to a group called SU(n) is written as su(n).

The dimension of a Lie algebra refers to how many independent directions or "generators" it has, much like how we might describe the size of a space.

Basic examples

A Lie algebra can be very simple. If a special operation called the Lie bracket always gives zero, the Lie algebra is called abelian. A straight line is automatically abelian because of how the Lie bracket works.

Another example uses matrices, which are like grids of numbers. If we take some matrices and define their Lie bracket as their "commutator" — the difference between multiplying them in one order and the other — we get a Lie algebra. This is useful for square matrices of a certain size, forming the general linear Lie algebra. This shows how Lie algebras help describe how things do not switch places in complex math.

Main article: Endomorphism ring

Main article: General linear group

Main article: Invertible

Main article: Determinant

Main article: Algebraic group

Definitions

Lie algebras are special math tools that help us understand symmetry and change. They are made from a vector space. A vector space is a group of objects you can add together or multiply by numbers.

Lie algebras also use an operation called the "Lie bracket." The Lie bracket combines two items from the space. It follows special rules that make studying these structures useful in many parts of math and physics.

The Lie bracket has unique features. For example, it is "alternating." This means putting an element with itself always gives zero. It also follows the "Jacobi identity." This is a special rule that helps keep things consistent when using many brackets. These features let Lie algebras model complicated systems where interactions are not simple or easy to switch around.

Examples

Matrix Lie algebras are a special kind of Lie algebra made from matrices. A matrix group is a set of invertible matrices that follow certain rules. The Lie algebra linked to this group consists of matrices that show small changes in the group. The Lie bracket in these algebras is often the commutator of matrices. This helps explain how these small changes work together.

Some important examples are:

• The special linear group has matrices with determinant 1. Its Lie algebra is made of matrices with trace 0.
• The orthogonal group keeps the length of vectors the same. Its Lie algebra has skew-symmetric matrices.
• The unitary group keeps vector lengths the same in complex space. Its Lie algebra has skew-hermitian matrices.

These examples show how Lie algebras can be built from different types of matrices. They also show the interesting patterns in this part of mathematics.

Representations

A Lie algebra is a special kind of math tool that helps us understand symmetry. It has a set of vectors with a special operation called the "Lie bracket." This operation shows how different vectors are related.

In physics, Lie algebras are very useful. They describe symmetries in the laws of nature. For example, they help us understand how objects turn in space. By studying these algebras, scientists can split hard problems into smaller parts to find answers.

Structure theory and classification

Lie algebras can be grouped and studied based on certain properties. This helps us understand more complex structures called Lie groups.

Lie algebras can be abelian, nilpotent, or solvable. An abelian Lie algebra has a "Lie bracket" that always equals zero. Nilpotent Lie algebras have a special series that eventually becomes zero. Solvable Lie algebras also have a series that ends in zero, but in a different way.

Some Lie algebras are simple, meaning they have no nontrivial ideals, and some are semisimple, which means they contain no solvable ideals except zero. These classifications help mathematicians understand the structure of Lie algebras better.

Relation to Lie groups

Main article: Lie group–Lie algebra correspondence

Lie algebras are special math tools that help us understand Lie groups. Lie groups are groups that also have smooth shapes. Each Lie group has a matching Lie algebra. You can think of the Lie algebra as the "tangent space" at the group's identity point.

For every Lie algebra, there is a Lie group that matches it. Sometimes more than one group can match the same algebra. This link makes it easier to study and sort Lie groups by looking at their Lie algebras first. This idea is very useful in many parts of math.

Real form and complexification

When we study Lie algebras, we sometimes use different types of numbers. A real form of a complex Lie algebra is a special version that uses real numbers. When we "complexify" this real form — by adding imaginary numbers — we get back to the original complex Lie algebra.

For example, the complex Lie algebra sl(2, C) has two different real forms: sl(2, R) and su(2). There are also special types of real forms. A split form is one where a certain part of the algebra behaves in a simple way. A compact form comes from the Lie algebra of a compact Lie group. Both split and compact forms are special and unique.

Main article: complex Lie algebra

Main articles: real form, split form, compact form

Lie algebra with additional structures

A Lie algebra can have extra features that work well with its main operation, the Lie bracket. One example is a graded Lie algebra, which has a special kind of organization called grading. Another example is a differential graded Lie algebra, which includes a tool called a differential that helps organize the vectors.

These ideas are used in advanced math, like studying the homotopy groups of certain spaces. A mathematician named Daniel Quillen used differential graded Lie algebras with rational numbers to explain rational homotopy theory in algebraic terms.

Lie ring

A Lie ring is a special kind of math idea that comes from studying groups. It is made from a group by looking at how parts of the group work together.

Lie rings help mathematicians learn about groups, especially those linked to prime numbers. They are important in areas like the study of p-adic numbers and group schemes. One example of a Lie ring comes from the dihedral group, which describes symmetries of regular shapes.

Definition using category-theoretic notation

The definition of a Lie algebra can also be described using a special area of mathematics called category theory. We think about the Lie algebra in terms of maps between spaces, not the individual elements inside those spaces.

We use two special kinds of maps, called "braiding isomorphisms", to help describe the Lie algebra. These maps let us rearrange the order of elements in a pattern. With these tools, a Lie algebra is defined as an object in the world of vector spaces, together with a special map that follows two important rules.

Generalization

Main article: Generalization of a Lie algebra

Mathematicians have found many ways to expand the idea of a Lie algebra. Many of these ideas come from physics. Some of these expansions include graded Lie algebras, Lie superalgebras, and Lie n-algebras. These help us learn about more complex structures and relationships in advanced mathematics and science.