Algebra over a field

If K is a field, then an algebra over K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication. A straightforward generalisation allows K to be any commutative ring. Oftentimes, the word algebra is used to refer to an associative algebra or even a unital associative algebra, but this is NOT the use of the word in this article.

Table of contents

1 Definitions 2 Properties 3 Kinds of algebras and examples 4 See also

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×A→A, with the result of this operation applied to the vectors x and y in A written as xy. Suppose further that:

• (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. If the multiplication is commutative, then A is called a commutative algebra.

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

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

Properties

Vector multiplication is a bilinear operator from A × A to A, and is therefore completely determined by the multiplication of basis elements of A. Conversely, once a basis for A has been choses, 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:

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

In mathematical physics, the structure coefficients are often written c_i,j^k, and their defining rule is written using the Einstein summation convention 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.

Kinds of algebras and examples

The most familiar kinds of algebras are those in which multiplication is associative.

• Examples of Associative algebras include:
  • 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 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.
  • 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.

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 (xy)z + (yz)x + (zx)y = 0 and also xy = -yx. For these algebras the product is called the Lie bracket and is 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 tangent vector fieldss 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)xx = x(yxx) 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)(ab + ba). 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 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.

• Quadradic 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, quadradic 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).

In geometric quantisation, one considers Poisson algebras, which carry two multiplications, turning them into commutative algebras and Lie algebras in different ways.

See also

• division algebra
• Clifford algebra
• Geometric algebra
• Incidence algebra