Algebra over a field

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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 K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring.

(Some authors use the term "algebra" synonymously with "associative algebra", but Wikipedia does not. Note also the other uses of the word listed in the algebra article.)

Definitions

To be precise, let K be a field, and let A be a vector space over K. Suppose we are given a binary operation A×AA, with the result of this operation applied to the vectors x and y in A written as xy. Suppose further that the operation is bilinear, i.e.:

  • (x + y)z = xz + yz;
  • x(y + z) = xy + xz;
  • (ax)y = a(xy); and
  • x(by) = b(xy);

for all scalars a and b in K and all vectors x, y, and z. Then with this operation, A becomes an algebra over K, and K is the base field of A. The operation is called "multiplication"; note the absence of associativity.

In general, xy is the product of x and y, and the operation is called multiplication. However, the operation in several special kinds of algebras goes by different names.

Algebras can also more generally be defined over any commutative ring K: we need a module A over K and a bilinear multiplication operation which satisfies the same identities as above; then A is a K-algebra, and K is the base ring of A.

Two algebras A and B over K are isomorphic if there exists a bijective K-linear map f : AB such that f(xy) = f(x) f(y) for all x,y in A. For all practical purposes, isomorphic algebras are identical; they just differ in the notation of their elements.

Properties

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 that the resulting multiplication will satisfy 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 ci,j,k, which are scalars. These structure coefficients determine the multiplication in A via the following rule:

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where e1,...,en form a basis of A. The only requirement on the structure coefficients is that, if the dimension n is an infinite number, 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 ci,jk, and their defining rule is written using the Einstein notation as

eiej = ci,jkek.

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

(xy)k = ci,jkxiyj.

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 generating set of 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.

Kinds of algebras and examples

A commutative algebra is one whose multiplication is commutative; an associative algebra is one whose multiplication is associative. These include the most familiar kinds of algebras.

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

  • Jordan algebras, for which we require (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, for which we 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 algebras are the reals, complexes, quaternions and octonions.
  • Power-associative algebras, for which we require that xmxn = xm+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.

More classes of algebras:

  • Division algebras, in which multiplicative inverses exist or division can be carried out. The finite-dimensional division algebras over the field of real numbers can be classified nicely.
  • Quadratic algebras, for which we require that xx = re + sx, for some elements r and s in the ground field, and e a unit for the algebra. Examples include all finite-dimensional alternative algebras, and the algebra of real 2-by-2 matrices. Up to isomorphism the only alternative, quadratic real algebras without divisors of zero are the reals, complexes, quaternions, and octonions.

Index-free notation

The definition of a k-algebra is also frequently given in index-free notation. In this case, a k-algebra is a field K and a ring A together with a ring morphism

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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

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given by

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K-algebra morphism

Given K-algebras A and B, a K-algebra morphism is a map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:A\to B} such that f is a ring morphism that commutes with the scalar multiplication defined by η; that is, the following diagram commutes:

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which one may write as

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for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\in K} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in A} . The space of all K-algebra morphisms is frequently written as

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A K-algebra isomorphism is a bijective K-algebra morphism.

See also

de:Algebra (Struktur) es:Álgebra sobre un cuerpo fr:Algèbre sur un corps it:Algebra su campo he:אלגברה לא אסוציאטיבית nl:Algebra (structuur) ja:多元環 pl:K-algebra