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 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: A → B 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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.

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.

• Associative algebras:
  • 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.

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

More classes of algebras:

• Graded algebras. These include most of the algebras of interest to multilinear algebra, such as the tensor algebra, symmetric algebra, and exterior algebra over a given vector space.

• Division algebras, in which multiplicative inverses exist or division can be carried out. The finite-dimensional alternative division algebras over the field of real numbers can be classified nicely. They are the real numbers (dimension 1), the complex numbers (dimension 2), the quaternions (dimension 4), and the octonions (dimension 8).

• 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.

• The Cayley-Dickson algebras (where K is R), which begin with:
  • C (a commutative and associative algebra);
  • the quaternions H (an associative algebra);
  • the octonions (an alternative algebra);
  • the sedenions (a power-associative algebra, like all of the Cayley-Dickson algebras).

• The Poisson algebras are considered in geometric quantization. They carry two multiplications, turning them into commutative algebras and Lie algebras in different ways.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): K\times A \to A

given by

Failed to parse (Missing texvc executable; please see math/README to configure.): (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

Failed to parse (Missing texvc executable; please see math/README to configure.): f(ka)=kf(a)

for all Failed to parse (Missing texvc executable; please see math/README to configure.): k\in K

and Failed to parse (Missing texvc executable; please see math/README to configure.): a \in A

. In other words, the following diagram commutes:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 c_i,j,k, which are scalars. These structure coefficients determine the multiplication in A via the following rule:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 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.

See also

• Clifford algebra
• Differential algebra
• Geometric algebra
• Max-plus algebrade:Algebra (Struktur)

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