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.

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:

<math>\mathbf{e}_{i} \mathbf{e}_{j} = \sum_{k=1}^n c_{i,j,k} \mathbf{e}_{k}</math>

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 important types of algebras are:

• Associative algebras, such as:
  • 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.
• Lie algebras, such as:
  • Euclidean space R³ with multiplication given by the vector cross product (with K the field R of real numbers);
  • algebras of tangent 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
• 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