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.

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 dimension of the associative algebra A over the field K is its dimension as a K-vector space.

Modules

The preceding definition generalizes without any change to an algebra over a commutative ring K. Such a space is then a module, rather than a vector space, over K with a bilinear form. A unital R-algebra A can equivalently be defined as a ring A with a ring homomorphism R→A. For instance:

• The n-by-n matrices with integer entries form a unital associative algebra over the integers.
• The polynomials with coefficients in the ring Z/nZ, the integers modulo n, form a unital associative algebra over Z/nZ.

See algebra (ring theory) for more.

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 if complex numbers are treated as a subset of the quaternions, complex numbers and quaternions don't commute).
• The real matrices (2 x 2) form an associative algebra useful in plane mapping.
• 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.

Associativity and the multiplication mapping

Associativity was defined above quantifying over all elements of A. It is possible to define associativity in a way that does not explicitly refer to elements. 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

$M\; circ\; (mbox\; \{Id\}\; times\; M)\; =\; M\; circ\; (M\; times\; mbox\; \{Id\})$

Here, the symbol $circ$ refers to function composition, and Id : A → A is the identity map on 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

$(M\; circ\; (mbox\; \{Id\}\; times\; M))\; (x,y,z)\; =\; M\; (x,\; M(y,z))$

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

$eta:\; K\; rightarrow\; A$

which has the property

$M\; circ\; (mbox\; \{Id\}\; times\; eta\; )\; =\; s\; =\; M\; circ\; (eta\; times\; mbox\; \{Id\})$

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: $s:Ktimes\; A\; rightarrow\; A$; thus, the above identity is sometimes written with Id standing in the place of s, with scalar multiplication being implicitly understood.

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 ρ: A → gl(V) from A to the general linear algebra of some vector space (or module) V that preserves the multiplicative operation: that is, ρ(xy)=ρ(x)ρ(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 $sigma:Arightarrow\; gl(V)$ and $tau:Arightarrow\; gl(W)$. 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

$rho(kx)\; =\; sigma(kx)\; otimes\; tau(kx)\; =\; ksigma(x)\; otimes\; ktau(x)\; =\; k^2\; (sigma(x)\; otimes\; tau(x))\; =\; k^2\; rho(x)$

for k ∈ K. One can rescue this attempt and restore linearity by imposing additional structure, by defining a map Δ: A → A × A, and defining the tensor product representation as

$rho\; =\; (sigmaotimes\; tau)\; circ\; Delta.$

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,

$x\; mapsto\; rho\; (x)\; =\; sigma(x)\; otimes\; mbox\{Id\}\_W\; +\; mbox\{Id\}\_V\; otimes\; tau(x)$

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:

$rho(xy)\; =\; sigma(x)\; sigma(y)\; otimes\; mbox\{Id\}\_W\; +\; mbox\{Id\}\_V\; otimes\; tau(x)\; tau(y)$.

But, in general, this does not equal

$rho(x)rho(y)\; =\; sigma(x)\; sigma(y)\; otimes\; mbox\{Id\}\_W\; +\; sigma(x)\; otimes\; tau(y)\; +\; sigma(y)\; otimes\; tau(x)\; +\; mbox\{Id\}\_V\; otimes\; tau(x)\; tau(y)$.

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

• Bourbaki, N. (1989). Algebra I. Springer.
• Ross Street, Quantum Groups: an entrée to modern algebra (1998). (Provides a good overview of index-free notation)