Algebra over a field

This article is about a particular kind of vector space. For other uses of the term "algebra", see algebra (disambiguation).

In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it is an algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as an algebra, this multiplication must satisfy certain compatibility axioms with the given vector space structure, such as distributivity. In other words, an algebra over a field is a set together with operations of multiplication, addition, and scalar multiplication by elements of the field.^[1]

One may generalize this notion by replacing the field of scalars by a commutative ring, and thus defining an algebra over a ring.

Note that while some authors may use the term "algebra" to describe a unital "associative algebra", this will not be the case for this article.

Definition and motivation

First example: The complex numbers

Any complex number may be written a + bi, where a and b are real numbers and i is the imaginary unit. In other words, a complex number is represented by the vector (a, b) over the field of real numbers. So the complex numbers form a two-dimensional real vector space, where addition is given by (a, b) + (c, d) = (a + c, b + d) and scalar multiplication is given by c(a, b) = (ca, cb), where all of a, b, c and d are real numbers. We use the symbol · to multiply two vectors together, which we use complex multiplication to define: (a, b) · (c, d) = (ac − bd, ad + bc).

The following statements are basic properties of the complex numbers. Let x, y, z be complex numbers, and let a, b be real numbers.

• (x + y) · z = (x · z) + (y · z). In other words, multiplying a complex number by the sum of two other complex numbers, is the same as multiplying by each number in the sum, and then adding.
• (ax) · (by) = (ab)(x · y). This shows that complex multiplication is compatible with the scalar multiplication by the real numbers.

This example fits into the following definition by taking the field K to be the real numbers, and the vector space A to be the complex numbers.

Definition

Let K be a field, and let A be a vector space over K equipped with an additional binary operation from A × A to A, denoted here by · (i.e. if x and y are any two elements of A, x · y is the product of x and y). Then A is an algebra over K if the following identities hold for any three elements x, y, and z of A, and all elements ("scalars") a and b of K:

• Left distributivity: (x + y) · z = x · z + y · z
• Right distributivity: x · (y + z) = x · y + x · z
• Compatibility with scalars: (ax) · (by) = (ab)(x · y).

These three axioms are another way of saying that the binary operation is bilinear. An algebra over K is sometimes also called a K-algebra, and K is called the base field of A. The binary operation is often referred to as multiplication in A. 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.

Notice that when a binary operation on a vector space is commutative, as in the above example of the complex numbers, it is left distributive exactly when it is right distributive. But in general, for non-commutative operations (such as the next example of the quaternions), they are not equivalent, and therefore require separate axioms.

A motivating example: quaternions

Main article: Quaternion

The real numbers may be viewed as a one dimensional vector space with a compatible multiplication, and hence a one dimensional algebra over itself. We saw above that the complex numbers form a two dimensional vector space over the field of real numbers, and hence form a two dimension algebra over the reals. In both these examples, every non-zero vector has an inverse. It is natural to ask whether one can similarly define a multiplication on a three dimensional real vector space such that every non-zero element has an inverse. The answer is no (see normed division algebras).

Nevertheless, in 1843, the quaternions were defined and provided the now famous four dimensional example of an algebra over the real numbers, where one can not only multiply vectors, but also divide. Any quaternion may be written as (a, b, c, d) = a + bi + cj + dk. Unlike the complex numbers, the quaternions are an example of a non-commutative algebra: for instance, (0,1,0,0) · (0,0,1,0) = (0,0,0,1) but (0,0,1,0) · (0,1,0,0) = (0,0,0,-1).

The quaternions were soon followed by several other hypercomplex number systems, which were the early examples of algebras over a field.

Basic concepts

Algebra homomorphisms

Main article: Algebra homomorphism

Given K-algebras A and B, a K-algebra homomorphism is a K-linear map f: A → B such that f(xy) = f(x) f(y) for all x,y in A. The space of all K-algebra homomorphisms is frequently written as

$\mathbf{Hom}_{K\text{-alg}} (A,B).$

A K-algebra isomorphism is a bijective K-algebra morphism. For all practical purposes, isomorphic algebras differ only by notation.

Subalgebras and ideals

Main article: Substructure

A subalgebra of an algebra over a field K, 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 subset of elements that is closed under addition, multiplication, and scalar multiplication. In symbols, we say that a subset L of a K-algebra A is a subalgebra if for every x, y in L and c in K, we have that x · y, x + y, and cx are all in L.

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 K-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 L of a K-algebra A is a left ideal if for every x and y in L, z in A and c in K, we have the following three statements.

• 1) x + y is in L (L is closed under addition),
• 2) cx is in L (L is closed under scalar multiplication),
• 3) z · x is in L (L is closed under left multiplication by arbitrary elements).

If (3) were replaced with x · z is in L, 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. Notice that conditions (1) and (2) together are equivalent to L being a linear subspace of A. It follows from condition (3) that every left or right ideal is a subalgebra.

It is important to notice that 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

Main article: Extension of scalars

If we have a field extension F/K, which is to say a bigger field F that contains K, then there is a natural way to construct an algebra over F from any algebra over K. It is the same construction one uses to make a vector space over a bigger field, namely the tensor product $V_F:=V \otimes_K F$. So if A is an algebra over K, then A_F is an algebra over F.

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.

Associative algebras

Main article: Associative algebra

• the algebra of all n-by-n matrices over the field (or commutative ring) K. 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 K[x] of all polynomials over K.
• algebras of functions, such as the R-algebra of all real-valued continuous functions defined on the interval [0,1], or the C-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 algebras

Main article: Non-associative algebra

A non-associative algebra^[2] (or distributive algebra) over a field K is a K-vector space A equipped with a K-bilinear map $A \times A \rightarrow A$. There are left and right multiplication maps $L_a : x \mapsto ax$ and $R_a : x \mapsto xa$. The enveloping algebra of A is the subalgebra of all K-endomorphisms of A generated by the multiplication maps.

An algebra is unital or unitary if it has a unit or identity element I with Ix = x = xI for all x in the algebra.

The best-known kinds of non-associative algebras are nearly associative—that is, some simple equation constrains the differences between different ways of associating multiplication of elements. These include:

• Lie algebras that are anticommutative , which require xx = 0 and the Jacobi identity (xy)z + (yz)x + (zx)y = 0. For these algebras the product is called the Lie bracket and is conventionally written [x,y] instead of xy. Examples include:
  • Euclidean space R³ with multiplication given by the vector cross product (with K the field R of real numbers);
  • algebras of vector fields on a differentiable manifold (if K is R or the complex numbers C) or an algebraic variety (for general K);
  • every associative algebra gives rise to a Lie algebra by using the commutator as Lie bracket. In fact every Lie algebra can either be constructed this way, or is a subalgebra of a Lie algebra so constructed.

• Jordan algebras that are commutative and satisfy the Jordan property (xy)x² = x(yx²) and also xy = yx.
  • every associative algebra over a field of characteristic other than 2 gives rise to a Jordan algebra by defining a new multiplication x*y = (1/2)(xy + yx). In contrast to the Lie algebra case, not every Jordan algebra can be constructed this way. Those that can are called special.

• Alternative algebras, which require that (xx)y = x(xy) and (yx)x = y(xx). The most important examples are the octonions (an algebra over the reals), and generalizations of the octonions over other fields. (Obviously all associative algebras are alternative.) Up to isomorphism the only finite-dimensional real alternative, division algebras (see below) are the reals, complexes, quaternions and octonions.

• Power-associative algebras, which require that x^mxⁿ = x^m+n, where m ≥ 1 and n ≥ 1. (Here we formally define xⁿ recursively as x(xⁿ⁻¹).) Examples include all associative algebras, all alternative algebras, and the sedenions.

These properties are related by: associative implies alternative, which in turn implies power associative; commutative and associative implies Jordan, which implies power associative. None of the converse implications hold.

Algebras and rings

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

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

where Z(A) is the center of A. Since η is a ring morphism, then one must have either that A is the trivial ring, or that η 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 K-algebras A and B, a unital K-algebra morphism f: A → B is a ring morphism that commutes with the scalar multiplication defined by η, which one may write as

f(ka) = kf(a)

for all $k\in K$ and $a \in A$. In other words, the following diagram commutes:

$\begin{matrix} && K && \\ & \eta_A \swarrow & \, & \eta_B \searrow & \\ A && \begin{matrix} f \\ \longrightarrow \end{matrix} && B \end{matrix}$

Structure coefficients

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

Thus, given the field K, any algebra can be specified up to isomorphism by giving its dimension (say n), and specifying n³ structure coefficients c_i,j,k, which are scalars. These structure coefficients determine the multiplication in A via the following rule:

$\mathbf{e}_{i} \mathbf{e}_{j} = \sum_{k=1}^n c_{i,j,k} \mathbf{e}_{k}$

where e₁,...,e_n form a basis of A. The only requirement on the structure coefficients is that, if the dimension n is infinite, then this sum must always converge (in whatever sense is appropriate for the situation).

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

When the algebra can be endowed with a metric, then the structure coefficients are 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, in mathematical physics, the structure coefficients are often written c_i,j^k, and their defining rule is written using the Einstein notation as

e_ie_j = c_i,j^ke_k.

If you apply this to vectors written in index notation, then this becomes

(xy)^k = c_i,j^kxⁱy^j.

If K is only a commutative ring and not a field, then the same process works if A is a free module over K. If it isn't, then the multiplication is still completely determined by its action on a set that spans A; 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 algebras

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.^[3]

There exist two two-dimensional algebras. Each algebra consists of linear combinations (with complex coefficients) of two basis elements, 1 (the unit) and a. According to the definition of a unit,

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

It remains to specify

$\textstyle a a = 1$   for the first algebra,

$\textstyle a a = 0$   for the second algebra.

There exist five three-dimensional algebras. Each algebra consists of linear combinations of three basis elements, 1 (the unit), a and b. Taking into account the definition of a unit, it is sufficient to specify

$\textstyle a a = a \, , \quad b b = b \, , \quad a b = b a = 0$   for the first algebra,

$\textstyle a a = a \, , \quad b b = 0 \, , \quad a b = b a = 0$   for the second algebra,

$\textstyle a a = b \, , \quad b b = 0 \, , \quad a b = b a = 0$   for the third algebra,

$\textstyle a a = 1 \, , \quad b b = 0 \, , \quad a b = - b a = b$   for the fourth algebra,

$\textstyle a a = 0 \, , \quad b b = 0 \, , \quad a b = b a = 0$   for the fifth algebra.

The fourth algebra is non-commutative, others are commutative.

See also

• Clifford algebra
• Differential algebra
• Geometric algebra
• Max-plus algebra

Notes

1. ^ See also Hazewinkel et. al. (2004), p. 3.
2. ^ Richard D. Schafer, An Introduction to Nonassociative Algebras (1996) ISBN 0-486-68813-5 Gutenberg eText
3. ^ Study, E. (1890), "Über Systeme complexer Zahlen und ihre Anwendungen in der Theorie der Transformationsgruppen", Monatshefte fũr Mathematik 1 (1): 283–354, doi:10.1007/BF01692479 

References

• Michiel Hazewinkel, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. Algebras, rings and modules. Volume 1. 2004. Springer, 2004. ISBN 1-4020-2690-0