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.
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 1
x =
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.
- 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.
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.
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/nZ (see modular arithmetic) form an associative algebra over Z/nZ.
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[?].
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