Associative algebra

In mathematics, an associative algebra is a vector space (or more generally module) which also allows the multiplication of vectors in a distributive and associative manner. They are thus special algebras.

Table of contents

1 Definition 2 Examples 3 Algebra homomorphisms 4 Generalizations 5 Coalgebras

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

If A contains an identity element, i.e. an element 1 such that 1x = x1 = x for all x in A, then we call A an associative algebra with one or a unitary (or unital) associative algebra. Such an algebra is a ring and contains a copy of the ground field K in the form {a1 : a in K}.

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 (but not an algebra over the complex numbers, since complex numbers don't commute with quaternions).
• 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 (using composition of operators as multiplication); 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; here we add and multiply functions pointwise.
• An example of a non-unitary associative algebra is given by the set of all functions f: R -> R whose limit as x nears infinity is zero.
• The Clifford algebras are useful in geometry and physics.
• Incidence algebras of locally finite partially ordered sets are unitary associative algebras considered in combinatorics.

Algebra homomorphisms

If A and B are associative algebras over the same field K, an algebra homomorphism h: A -> B is a K-linear map which is also multiplicative in the sense that h(xy) = h(x) h(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 f the number f(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. In this case, a unitary R-algebra A can equivalently be defined as a ring A with a ring homomorphism R→A.

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

Coalgebras

An associative unitary algebra over K is based on a morphism A×A→A having 2 inputs (multiplicator and multiplicand) and one output (product), as well as a morphism K→A identifying the scalar multiples of the multiplicative identity. These two morphisms can be dualized using categorial duality by reversing all arrows in the commutative diagrams which describe the algebra axioms; this defines the structure of a coalgebra.