# Associative algebra

*This article is about a particular kind of vector space. For other uses of the term "algebra" see algebra (disambiguation).*

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

## Contents

## 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 1*x* = *x*1 = *x* for all *x* in *A*, then we call *A* an *associative algebra with one* or a **unital** (or **unitary**) **associative algebra**.
Such an algebra is a ring, and contains all elements *a* of the field *K* by identification with *a*1.

The preceding definition generalizes without any change to an algebra over a commutative ring *K* (except that a *K*-linear space is then called a module and not a vector space). See algebra (ring theory) for more.

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*.

## Index-free notation

In the above definition of an associative algebra, the definition of associativity was made with regard to all of the elements of *A*. It is sometimes more convenient to have a definition of associativity that does not need to refer to the elements of *A*.
This can be done as follows. An algebra is defined as a map *M* (multiplication) on a vector space *A*:

An associative algebra is an algebra where the map *M* has the property

Here, the symbol refers to functional composition, and Id is the identity map: for all *x* in *A*. To see the equivalence of the definitions, we need only understand that each side of the above equation is a function that takes three arguments. For example, the left-hand side acts as

Similarly, a unital associative algebra can be defined in terms of a unit map

which has the property

Here, the unit map η takes an element *k* in *K* to the element *k1* in *A*, where *1* is the unit element of *A*. The map *s* is just plain-old scalar multiplication: ; thus, the above identity is sometimes written with Id standing in the place of *s*, with scalar multiplication being implicitly understood.

## 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**/*n***Z** (see modular arithmetic) form an associative algebra over **Z**/*n***Z**.

## 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.

There is also an abstract notion of F-coalgebra.

## Representations

A representation of an algebra is a linear map from *A* to the general linear algebra of some vector space (or module) *V* that preserves the multiplicative operation: that is, .

Note, however, that there is no natural way of defining a tensor product of representations of associative algebras, without somehow imposing additional conditions. Here, by *tensor product of representations*, the usual meaning is intended: the result should be a linear representation on the product vector space.
Imposing such additional structure typically leads to the idea of a Hopf algebra or a Lie algebra, as demonstrated below.

### Motivation for a Hopf algebra

Consider, for example, two representations and . One might try to form a tensor product representation according to how it acts on the product vector space, so that

- .

However, such a map would not be linear, since one would have

for . One can rescue this attempt and restore linearity by imposing additional structure, by defining a map , and defining the tensor product representation as

- .

Here, Δ is a coalgebra. The resulting structure is called a bialgebra. To be consistent with the definitions of the associative algebra, the coalgebra must be co-associative, and, if the algebra is unital, then the co-algebra must be unital as well. Note that bialgebras leave multiplication and co-multiplication unrelated; thus it is common to relate the two (by defining an antipode), thus creating a Hopf algebra.

### Motivation for a Lie algebra

One can try to be more clever in defining a tensor product. Consider, for example,

so that the action on the tensor product space is given by

- .

This map is clearly linear in *x*, and so it does not have the problem of the earlier definition. However, it fails to preserve multiplication:

- .

But, in general, this does not equal

- .

Equality would hold if the product *xy* were antisymmetric (if the product were the Lie bracket, that is, ), thus turning the associative algebra into a Lie algebra.

## References

- Ross Street,
*Quantum Groups: an entrée to modern algebra*(1998).*(Provides a good overview of index-free notation)*

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