Home > 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.

1 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.

2 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 polynomialsIn mathematics polynomial functions or polynomials are an important class of simple and smooth functions. Simple means they are constructed using only multiplication and addition. Smooth means they are infinitely differentiable, i. they have derivatives o with real coefficients form a unitary associative algebra over the reals.
• Given any Banach spaceFunctional analysis In mathematics, Banach spaces named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. Banach spaces are typically infinite-dimensional spaces containing functions. Definition Banach s 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 algebraIn functional analysis, a Banach algebra named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. The algebra multiplication and the Banach space norm are required to be related.
• Given any topological spaceTopology is the study or science of places. It derives its name from the Greek words τοπος meaning place and λογος meaning study, talk. See also earth science, geography, human geography, g 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 algebraClifford algebras are a type of associative algebra in mathematics. They can be thought of as generalizations of the complex numbers and quaternions. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonals are useful in geometry and physics.
• Incidence algebras of locally finite partially ordered sets are unitary associative algebras considered in combinatorics.