SafekipediaIn mathematics, an algebra over a field (often simply called an algebra) is a special kind of mathematical structure. It is a vector space that also has a way to multiply its elements together in a consistent way. This multiplication follows certain rules, making the algebra a useful tool in many areas of math.
One important feature of an algebra is that its multiplication might or might not follow the associative property. When it does, it is called an associative algebra. For example, the collection of real square matrices of a certain size forms an associative algebra over the field of real numbers. These matrices can be added and multiplied together in ways that keep the structure consistent.
Algebras can also have a special element called an identity element, which acts like the number 1 in regular multiplication. When an algebra has such an element, it is called unital or unitary. The ring of real square matrices has this property because there is a special matrix, the identity matrix, that leaves other matrices unchanged when multiplied by them. This makes it a unital associative algebra, which means it is both associative and has an identity element.
Definition and motivation
An algebra over a field is a special kind of mathematical space that combines ideas from vector spaces and multiplication. Imagine a vector space, which is a collection of objects that can be added together and stretched or shrunk by numbers (called scalars). Now, add a multiplication rule that tells us how to combine any two objects in this space to get another object.
This multiplication must follow certain rules, like distributing over addition — similar to how we multiply numbers in regular arithmetic. For example, if you multiply a sum of two objects by another object, it should be the same as multiplying each object in the sum separately and then adding the results. Algebras can be associative, where the order of multiplication doesn’t matter, or non-associative, where it does.
| Algebra | vector space | bilinear operator | associativity | commutativity |
|---|---|---|---|---|
| complex numbers | R 2 {\displaystyle \mathbb {R} ^{2}} | product of complex numbers ( a + i b ) ⋅ ( c + i d ) {\displaystyle \left(a+ib\right)\cdot \left(c+id\right)} | yes | yes |
| cross product of 3D vectors | R 3 {\displaystyle \mathbb {R} ^{3}} | cross product a → × b → {\displaystyle {\vec {a}}\times {\vec {b}}} | no | no (anticommutative) |
| quaternions | R 4 {\displaystyle \mathbb {R} ^{4}} | Hamilton product ( a + v → ) ( b + w → ) {\displaystyle (a+{\vec {v}})(b+{\vec {w}})} | yes | no |
| polynomials | R [ X ] {\displaystyle \mathbb {R} [X]} | polynomial multiplication | yes | yes |
| square matrices | R n × n {\displaystyle \mathbb {R} ^{n\times n}} | matrix multiplication | yes | no |
Basic concepts
An algebra over a field is a special kind of mathematical space that has two main operations: addition and multiplication. It also allows multiplying by numbers from a field, which means we can stretch or shrink the space in certain ways.
We can study smaller parts of an algebra called subalgebras. These are sets of elements inside the algebra that stay closed under addition, multiplication, and stretching by numbers. We can also look at special subsets called ideals, which behave in particular ways when multiplied by other elements in the algebra.
Kinds of algebras and examples
Algebras over fields can be many different types, depending on extra rules we add, like whether their multiplication is commutative or associative. These rules change how we study them.
One special type is a unital algebra, which has a special element that acts like a "1", so when you multiply anything by this element, you get the same thing back. Another type is a zero algebra, where multiplying any two elements always gives zero.
An associative algebra follows the usual order of operations when multiplying. Examples include matrices and polynomials.
A non-associative algebra doesn’t need to follow that order. Examples include special kinds of numbers like octonions and structures like Euclidean space.
Main article: Associative algebra
Main article: Non-associative algebra
Algebras and rings
An algebra over a field can also be described as a special type of ring. This ring has a special connection to the field through a map that links the field's elements to the ring in a consistent way. This connection ensures that the algebra behaves nicely with respect to both the ring operations and the field's scalar multiplication.
When we have two such algebras, a special kind of map between them, called a homomorphism, preserves both the ring structure and how the field acts on the algebra. This means the map respects the way elements are multiplied and added together in both algebras.
Structure coefficients
Main article: Structure constants
In algebras over a field, the way elements multiply together is completely decided by how the basic building blocks, or basis elements, multiply. Once we pick these building blocks, we can freely decide how they multiply, and this decision uniquely determines how all other elements multiply.
For a finite-dimensional algebra, this multiplication is captured by special numbers called structure coefficients. If the algebra has dimension n, we need n3 such coefficients, written ci,j,k, to fully describe the multiplication. These coefficients follow a specific rule that tells us how the basis elements combine to form other elements in the algebra.
Classification of low-dimensional unital associative algebras over the complex numbers
Two-dimensional, three-dimensional, and four-dimensional algebras over complex numbers have been fully classified. In two dimensions, there are two types of algebras. Each uses two basic elements: 1 (which acts like the number 1) and another element called a.
For three-dimensional algebras, there are five different types. Each uses three basic elements: 1, a, and b. The way these elements multiply together determines which type of algebra it is. One of these five types does not follow the usual order of multiplication, while the others do.
Generalization: algebra over a ring
In some parts of mathematics, like commutative algebra, we talk about something called an algebra over a ring. Instead of using a field, we use a special kind of mathematical object called a commutative ring. The main idea stays the same, but we change one part: we think of the algebra as a special kind of structure called an R-module instead of a vector space.
A ring can always be seen as an associative algebra over its center and over the integers. For example, the split-biquaternion algebra is a type of algebra over its center, which isn’t a field. In commutative algebra, if we have a commutative ring, we can give it a structure that makes it an R-algebra using a special map from R to A. Not every ring can be turned into an algebra over a field, like the integers.
Main article: Associative algebra
This article is a child-friendly adaptation of the Wikipedia article on Algebra over a field, available under CC BY-SA 4.0.