Associative algebra

An associative algebra is a vector space which also allows the multiplication of vectors in a distributive and associative manner.

Definition

An associative algebra A over a field K is defined to be a vector space over K together with a K-bilinear multiplication A x A -> A (where the image of (x,y) is written as xy) such that the associativity law holds:

• (x y) z = x (y z) for all x, y and z in A.

The bilinearity of the multiplication can be expressed as

• (x + y) z = x z + y z    for all x, y, z in A,
• x (y + z) = x y + x z    for all x, y, z in A,
• α (x y) = (α x) y = x (α y)    for all x, y in A and α in K.

If A contains an identity element, i.e. an element 1 such that 1x = x1 = x for all x in K, then we call A an associative algebra with one or a unitary associative algebra. Such an algebra is a ring.

The dimension of the associative algebra A over the field K is its dimension as a K-vector space.

Examples

• The square n-by-n matrices with entries from the field K form a unitary associative algebra over K.
• The complex numbers form a 2-dimensional unitary associative algebra over the real numbers
• The quaternions form a 4-dimensional unitary associative algebra over the reals and a 2-dimensional unitary associative algebra over the complex numbers.
• The polynomials with real coefficients form a unitary associative algebra over the reals.
• Given any Banach space X, the continuous linear operators A : X -> X form a unitary associative algebra; this is in fact a Banach algebra.
• Given any topological space X, the continuous real- (or complex-) valued functions on X form a real (or complex) unitary associative algebra.
• An example of a non-unitary associative algebra is given by the functions f : R -> R whose limit for x→∞ is zero.

Algebra homomorphisms

If A and B are associative algebras over the same field K, an algebra homomorphism f : A -> B is a K-linear map which is also multiplicative in the sense that f(xy) = f(x) f(y) for all x, y in A. With this notion of morphism, the class of all associative algebras over K becomes a category.

Take for example the algebra A of all real-valued continuous functions R -> R, and B = R. Both are algebras over R, and the map which assigns to every continuous function φ the number φ(0) is an algebra homomorphism from A to B.

Generalizations

One may consider associative algebras over a commutative ring R: these are modules over R together with a R-bilinear map which yields an associative multiplication.

The n-by-n matrices with integer entries form an associative algebra over the integers and the polynomials with coefficients in the ring Z_n (see modular arithmetic) form an associative algebra over Z_n.