Associative algebra

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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

1 Definition
2 Examples
3 Algebra homomorphisms
4 Index-free notation
5 Generalizations
6 Coalgebras
7 Representations

7.1 Motivation for a Hopf algebra
7.2 Motivation for a Lie algebra

8 References

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 associative 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 unital (or unitary) associative algebra. Such an algebra is a ring, and contains all elements a of the field K by identification with a1.

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:

[M: A \times A \rightarrow A]

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

[M \circ (\mbox \times M) = M \circ (M \times \mbox )]

Here, the symbol [\circ] refers to functional composition, and Id is the identity map: [Id(x)=x] 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

[( M \circ (\mbox \times M)) (x,y,z) = M (x, M(y,z))]

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

[\eta: K \rightarrow A]

which has the property

[M \circ (\mbox \times \eta ) = s = M \circ (\eta \times \mbox )]

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: [s:K\times A \rightarrow A]; 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 unital 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.

There is also an abstract notion of F-coalgebra.

Representations

A representation of an algebra is a linear map [\rho:A\rightarrow gl(V)] from A to the general linear algebra of some vector space (or module) V that preserves the multiplicative operation: that is, [\rho(xy)=\rho(x)\rho(y)].

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 [\sigma:A\rightarrow gl(V)] and [\tau:A\rightarrow gl(W)]. One might try to form a tensor product representation [\rho: x \mapsto \rho(x) = \sigma(x) \otimes \tau(x)] according to how it acts on the product vector space, so that

[\rho(x)(v \otimes w) = (\sigma(x)(v)) \otimes (\tau(x)(w))].

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

[\rho(kx) = \sigma(kx) \otimes \tau(kx) = k\sigma(x) \otimes k\tau(x) = k^2 (\sigma(x) \otimes \tau(x)) = k^2 \rho(x)]

for [k \in K]. One can rescue this attempt and restore linearity by imposing additional structure, by defining a map [\Delta:A \rightarrow A \times A], and defining the tensor product representation as

[\rho = (\sigma\otimes \tau) \circ \Delta].

Here, Δ is a comultiplication. 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,

[x \mapsto \rho (x) = \sigma(x) \otimes \mbox_W + \mbox_V \otimes \tau(x)]

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

[\rho(x) (v \otimes w) = (\sigma(x) v)\otimes w + v \otimes (\tau(x) w)].

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:

[\rho(xy) = \sigma(x) \sigma(y) \otimes \mbox_W + \mbox_V \otimes \tau(x) \tau(y)].

But, in general, this does not equal

[\rho(x)\rho(y) = \sigma(x) \sigma(y) \otimes \mbox_W + \sigma(x) \otimes \tau(y) +\sigma(y) \otimes \tau(x) +\mbox_V \otimes \tau(x) \tau(y)].

Equality would hold if the product xy were antisymmetric (if the product were the Lie bracket, that is, [xy \equiv M(x,y) = [x,y]]), 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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