Algebra over a field

In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".

The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras where associativity of multiplication is assumed, and non-associative algebras, where associativity is not assumed (but not excluded, either). Given an integer Template:Math, the ring of real square matrices of order Template:Math is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity instead.

An algebra is unital or unitary if it has an identity element with respect to the multiplication. The ring of real square matrices of order Template:Math forms a unital algebra since the identity matrix of order Template:Math is the identity element with respect to matrix multiplication. It is an example of a unital associative algebra, a (unital) ring that is also a vector space.

Many authors use the term algebra to mean associative algebra, or unital associative algebra, or in some subjects such as algebraic geometry, unital associative commutative algebra.

Replacing the field of scalars by a commutative ring leads to the more general notion of an algebra over a ring. Algebras are not to be confused with vector spaces equipped with a bilinear form, like inner product spaces, as, for such a space, the result of a product is not in the space, but rather in the field of coefficients.

Definition and motivation

Motivating examples

Algebra	vector space	bilinear operator	associativity	commutativity
complex numbers	${ℝ}^2$	product of complex numbersTemplate:Br$(a+ib)\cdot (c+id)$	yes	yes
cross product of 3D vectors	${ℝ}^3$	cross product Template:Br$\overrightarrow{a}\times \overrightarrow{b}$	no	no (anticommutative)
quaternions	${ℝ}^4$	Hamilton product Template:Br$(a+\overrightarrow{v})(b+\overrightarrow{w})$	yes	no
polynomials	$ℝ[X]$	polynomial multiplication	yes	yes
square matrices	${ℝ}^{n\times n}$	matrix multiplication	yes	no

Definition

Let Template:Math be a field, and let Template:Mvar be a vector space over Template:Mvar equipped with an additional binary operation from Template:Math to Template:Mvar, denoted here by Template:Math (that is, if Template:Mvar and Template:Mvar are any two elements of Template:Mvar, then Template:Math is an element of Template:Mvar that is called the product of Template:Mvar and Template:Mvar). Then Template:Mvar is an algebra over Template:Mvar if the following identities hold for all elements Template:Math in Template:Mvar, and all elements (often called scalars) Template:Mvar and Template:Mvar in Template:Mvar:

• Right distributivity: Template:Math
• Left distributivity: Template:Math
• Compatibility with scalars: Template:Math.

These three conditions are another way of saying that the binary operation is bilinear. An algebra over Template:Math is sometimes also called a Template:Mvar-algebra, and Template:Math is called the base field of Template:Math. The binary operation is often referred to as multiplication in Template:Math. The convention adopted in this article is that multiplication of elements of an algebra is not necessarily associative, although some authors use the term algebra to refer to an associative algebra.

When a binary operation on a vector space is commutative, left distributivity and right distributivity are equivalent, and, in this case, only one distributivity requires a proof. In general, for non-commutative operations left distributivity and right distributivity are not equivalent, and require separate proofs.

Basic concepts

Algebra homomorphisms

Given Template:Math-algebras Template:Math and Template:Math, a homomorphism of Template:Math-algebras or Template:Math-algebra homomorphism is a Template:Math-linear map Template:Math such that Template:Math for all Template:Math in Template:Math. If Template:Math and Template:Math are unital, then a homomorphism satisfying Template:Math is said to be a unital homomorphism. The space of all Template:Math-algebra homomorphisms between Template:Math and Template:Math is frequently written as

${𝐇𝐨𝐦}_{K\text{-alg}}(A,B).$

A Template:Math-algebra isomorphism is a bijective Template:Math-algebra homomorphism.

Subalgebras and ideals

Template:Main A subalgebra of an algebra over a field Template:Math is a linear subspace that has the property that the product of any two of its elements is again in the subspace. In other words, a subalgebra of an algebra is a non-empty subset of elements that is closed under addition, multiplication, and scalar multiplication. In symbols, we say that a subset Template:Math of a Template:Math-algebra Template:Math is a subalgebra if for every Template:Math, Template:Math in Template:Math and Template:Math in Template:Math, we have that Template:Math, Template:Math, and Template:Math are all in Template:Math.

In the above example of the complex numbers viewed as a two-dimensional algebra over the real numbers, the one-dimensional real line is a subalgebra.

A left ideal of a Template:Math-algebra is a linear subspace that has the property that any element of the subspace multiplied on the left by any element of the algebra produces an element of the subspace. In symbols, we say that a subset Template:Math of a Template:Math-algebra Template:Math is a left ideal if for every Template:Math and Template:Math in Template:Math, Template:Math in Template:Math and Template:Math in Template:Math, we have the following three statements.

1. Template:Math is in Template:Math (Template:Math is closed under addition),
2. Template:Math is in Template:Math (Template:Math is closed under scalar multiplication),
3. Template:Math is in Template:Math (Template:Math is closed under left multiplication by arbitrary elements).

If (3) were replaced with Template:Math is in Template:Math, then this would define a right ideal. A two-sided ideal is a subset that is both a left and a right ideal. The term ideal on its own is usually taken to mean a two-sided ideal. Of course when the algebra is commutative, then all of these notions of ideal are equivalent. Conditions (1) and (2) together are equivalent to Template:Math being a linear subspace of Template:Math. It follows from condition (3) that every left or right ideal is a subalgebra.

This definition is different from the definition of an ideal of a ring, in that here we require the condition (2). Of course if the algebra is unital, then condition (3) implies condition (2).

Extension of scalars

Template:Main If we have a field extension Template:Math, which is to say a bigger field Template:Math that contains Template:Math, then there is a natural way to construct an algebra over Template:Math from any algebra over Template:Math. It is the same construction one uses to make a vector space over a bigger field, namely the tensor product Template:Math. So if Template:Math is an algebra over Template:Math, then Template:Math is an algebra over Template:Math.

Kinds of algebras and examples

Algebras over fields come in many different types. These types are specified by insisting on some further axioms, such as commutativity or associativity of the multiplication operation, which are not required in the broad definition of an algebra. The theories corresponding to the different types of algebras are often very different.

Unital algebra

An algebra is unital or unitary if it has a unity or identity element Template:Math with Template:Math for all Template:Math in the algebra.

Zero algebra

An algebra is called a zero algebra if Template:Math for all Template:Math, Template:Math in the algebra, not to be confused with the algebra with one element. It is inherently non-unital (except in the case of only one element), associative and commutative.

A unital zero algebra is the direct sum Template:Math of a field Template:Math and a Template:Math-vector space Template:Math, that is equipped by the only multiplication that is zero on the vector space (or module), and makes it an unital algebra.

More precisely, every element of the algebra may be uniquely written as Template:Math with Template:Math and Template:Math, and the product is the only bilinear operation such that Template:Math for every Template:Math and Template:Math in Template:Math. So, if Template:Math and Template:Math, one has $$(k_1+v_1)(k_2+v_2)=k_1{k}_2+(k_1{v}_2+k_2{v}_1).$$

A classical example of unital zero algebra is the algebra of dual numbers, the unital zero Template:Math-algebra built from a one dimensional real vector space.

This definition extends verbatim to the definition of a unital zero algebra over a commutative ring, with the replacement of "field" and "vector space" with "commutative ring" and "module".

Unital zero algebras allow the unification of the theory of submodules of a given module and the theory of ideals of a unital algebra. Indeed, the submodules of a module Template:Math correspond exactly to the ideals of Template:Math that are contained in Template:Math.

For example, the theory of Gröbner bases was introduced by Bruno Buchberger for ideals in a polynomial ring Template:Math over a field. The construction of the unital zero algebra over a free Template:Math-module allows extending this theory as a Gröbner basis theory for submodules of a free module. This extension allows, for computing a Gröbner basis of a submodule, to use, without any modification, any algorithm and any software for computing Gröbner bases of ideals.

Similarly, unital zero algebras allow to deduce straightforwardly the Lasker–Noether theorem for modules (over a commutative ring) from the original Lasker–Noether theorem for ideals.

Associative algebra

Template:Main Examples of associative algebras include

• the algebra of all Template:Math-by-Template:Math matrices over a field (or commutative ring) Template:Math. Here the multiplication is ordinary matrix multiplication.
• group algebras, where a group serves as a basis of the vector space and algebra multiplication extends group multiplication.
• the commutative algebra Template:Math of all polynomials over Template:Math (see polynomial ring).
• algebras of functions, such as the Template:Math-algebra of all real-valued continuous functions defined on the interval Template:Math, or the Template:Math-algebra of all holomorphic functions defined on some fixed open set in the complex plane. These are also commutative.
• Incidence algebras are built on certain partially ordered sets.
• algebras of linear operators, for example on a Hilbert space. Here the algebra multiplication is given by the composition of operators. These algebras also carry a topology; many of them are defined on an underlying Banach space, which turns them into Banach algebras. If an involution is given as well, we obtain B*-algebras and C*-algebras. These are studied in functional analysis.

Non-associative algebra

A non-associative algebra (or distributive algebra) over a field Template:Math is a Template:Math-vector space A equipped with a Template:Math-bilinear map Template:Math. The usage of "non-associative" here is meant to convey that associativity is not assumed, but it does not mean it is prohibited – that is, it means "not necessarily associative".

Examples detailed in the main article include:

• Euclidean space Template:Math with multiplication given by the vector cross product
• Octonions
• Lie algebras
• Jordan algebras
• Alternative algebras
• Flexible algebras
• Power-associative algebras

Algebras and rings

The definition of an associative Template:Math-algebra with unit is also frequently given in an alternative way. In this case, an algebra over a field Template:Math is a ring Template:Math together with a ring homomorphism

$\eta :K\to Z(A),$

where Template:Math is the center of Template:Math. Since Template:Math is a ring homomorphism, then one must have either that Template:Math is the zero ring, or that Template:Math is injective. This definition is equivalent to that above, with scalar multiplication

$K\times A\to A$

given by

$(k,a)\mapsto \eta (k)a.$

Given two such associative unital Template:Math-algebras Template:Math and Template:Math, a unital Template:Math-algebra homomorphism Template:Math is a ring homomorphism that commutes with the scalar multiplication defined by Template:Math, which one may write as

$f(ka)=kf(a)$

for all Template:Math and Template:Math. In other words, the following diagram commutes:

$\begin{array}{ccccc}& & K& & \\ & {\eta }_A\swarrow & & {\eta }_B\searrow & \\ A& & \begin{array}{c}f\\ ⟶\end{array}& & B\end{array}$

Structure coefficients

Template:Main For algebras over a field, the bilinear multiplication from Template:Math to Template:Math is completely determined by the multiplication of basis elements of Template:Math. Conversely, once a basis for Template:Math has been chosen, the products of basis elements can be set arbitrarily, and then extended in a unique way to a bilinear operator on Template:Math, i.e., so the resulting multiplication satisfies the algebra laws.

Thus, given the field Template:Math, any finite-dimensional algebra can be specified up to isomorphism by giving its dimension (say Template:Math), and specifying Template:Math structure coefficients Template:Math, which are scalars. These structure coefficients determine the multiplication in Template:Math via the following rule:

${𝐞}_i{𝐞}_j=\sum _{k=1}^n{c}_{i,j,k}{𝐞}_k$

where Template:Math, ..., Template:Math form a basis of Template:Math.

Note however that several different sets of structure coefficients can give rise to isomorphic algebras.

In mathematical physics, the structure coefficients are generally written with upper and lower indices, so as to distinguish their transformation properties under coordinate transformations. Specifically, lower indices are covariant indices, and transform via pullbacks, while upper indices are contravariant, transforming under pushforwards. Thus, the structure coefficients are often written Template:Math, and their defining rule is written using the Einstein notation as

Template:Math.

Applying this to vectors written in index notation, then this becomes

Template:Math.

If Template:Math is only a commutative ring and not a field, then the same process works if Template:Math is a free module over Template:Math. If it isn't, then the multiplication is still completely determined by its action on a set that spans Template:Math; however, the structure constants can't be specified arbitrarily in this case, and knowing only the structure constants does not specify the algebra up to isomorphism.

Classification of low-dimensional unital associative algebras over the complex numbers

Two-dimensional, three-dimensional and four-dimensional unital associative algebras over the field of complex numbers were completely classified up to isomorphism by Eduard Study.

There exist two such two-dimensional algebras. Each algebra consists of linear combinations (with complex coefficients) of two basis elements, Template:Math (the identity element) and Template:Math. According to the definition of an identity element,

$1\cdot 1=1,1\cdot a=a,a\cdot 1=a.$

It remains to specify

$aa=1$   for the first algebra,

$aa=0$   for the second algebra.

There exist five such three-dimensional algebras. Each algebra consists of linear combinations of three basis elements, Template:Math (the identity element), Template:Math and Template:Math. Taking into account the definition of an identity element, it is sufficient to specify

$aa=a,bb=b,ab=ba=0$   for the first algebra,

$aa=a,bb=0,ab=ba=0$   for the second algebra,

$aa=b,bb=0,ab=ba=0$   for the third algebra,

$aa=1,bb=0,ab=-ba=b$   for the fourth algebra,

$aa=0,bb=0,ab=ba=0$   for the fifth algebra.

The fourth of these algebras is non-commutative, and the others are commutative.

Generalization: algebra over a ring

In some areas of mathematics, such as commutative algebra, it is common to consider the more general concept of an algebra over a ring, where a commutative ring Template:Math replaces the field Template:Math. The only part of the definition that changes is that Template:Math is assumed to be an [[Module (mathematics)|Template:Math-module]] (instead of a Template:Math-vector space).

Associative algebras over rings

A ring Template:Math is always an associative algebra over its center, and over the integers. A classical example of an algebra over its center is the split-biquaternion algebra, which is isomorphic to Template:Math, the direct product of two quaternion algebras. The center of that ring is Template:Math, and hence it has the structure of an algebra over its center, which is not a field. Note that the split-biquaternion algebra is also naturally an 8-dimensional Template:Math-algebra.

In commutative algebra, if Template:Math is a commutative ring, then any unital ring homomorphism Template:Math defines an Template:Math-module structure on Template:Math, and this is what is known as the Template:Math-algebra structure. So a ring comes with a natural Template:Math-module structure, since one can take the unique homomorphism Template:Math. On the other hand, not all rings can be given the structure of an algebra over a field (for example the integers). See Field with one element for a description of an attempt to give to every ring a structure that behaves like an algebra over a field.

See also

• Algebra over an operad
• Alternative algebra
• Clifford algebra
• Composition algebra
• Differential algebra
• Free algebra
• Geometric algebra
• Max-plus algebra
• Mutation (algebra)
• Operator algebra
• Zariski's lemma

References

• Template:Cite book