Definitions

Ring morphism

Algebra over a field

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 this article does not. Note also the other uses of the word listed in the algebra article.)

Definitions

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 juxtaposition (i.e. if x and y are any two elements of A, xy is the product of x and y). Then if the binary operation is bilinear, which means that the following identities hold for any three elements x, y, and z of A, and all elements ("scalars") a and b of K:

  • (x + y)z = xz + yz
  • x(y + z) = xy + xz
  • (ax)(by) = (ab)(xy)

we call A an algebra over K, we say that A is a K-algebra, and K is the base field of A. The binary operation is often referred to as multiplication in A. According to the convention adopted in this article (see above), multiplication of elements of A is not necessarily associative.

More generally, algebras can be defined over an arbitrary commutative ring K instead of a field. In this case A forms a K-module, with bilinear multiplication again satisfying the above identities. In this case, A is a K-algebra, and K is the base ring of A.

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

mathbf{Hom}_{Ktext{-alg}} (A,B).
A K-algebra isomorphism is a bijective K-algebra morphism. For all practical purposes, isomorphic algebras differ only by notation.

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:

  • Lie algebras, for which we 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:
  • 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, division algebras (see below) 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.
  • The hyperbolic quaternion algebra over R, which was an experimental algebra before the adoption of Minkowski space for special relativity.

More classes of algebras:

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

etacolon Kto 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

Ktimes 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: AB is a ring morphism that commutes with the scalar multiplication defined by η, which one may write as
f(ka)=kf(a)
for all kin 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 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:

mathbf{e}_{i} mathbf{e}_{j} = sum_{k=1}^n c_{i,j,k} mathbf{e}_{k}
where e1,...,en 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 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 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.

See also

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