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 (mathematics)) which also allows the multiplication of vectors in a distributivity and associativity manner. They are thus special algebra over a field.
= Definition =
An associative algebra A over a field (mathematics) K is defined to be a vector space over K together with a K -bilinear operator A x A → A (where the image of ( x , y ) is written as xy ) such that the associativity law holds:
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 (mathematics) and not a vector space). See algebra (ring theory) for more.
The dimension of the associative algebra A over the field K is its Hamel dimension as a K -vector space.
= Examples =
= Algebra homomorphisms =
If A and B are associative algebras over the same field K , an algebra homomorphism h : A → B is a K -linear transformation 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 theory.
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 imes A ightarrow A An associative algebra is an algebra where the map M has the property :M circ (mbox {Id} imes M) = M circ (M imes mbox {Id}) 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 {Id} imes 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 ightarrow A which has the property :M circ (mbox {Id} imes eta ) = s = M circ (eta imes 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:K imes A ightarrow 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 module (mathematics) 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 (algebra) 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 group representation of an algebra is a linear map ho:A ightarrow gl(V) from A to the general linear algebra of some vector space (or module) V that preserves the multiplicative operation: that is, ho(xy)= ho(x) ho(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 ightarrow gl(V) and au:A ightarrow gl(W). One might try to form a tensor product representation ho: x mapsto ho(x) = sigma(x) otimes au(x) according to how it acts on the product vector space, so that
: ho(x)(v otimes w) = (sigma(x)(v)) otimes ( au(x)(w)).
However, such a map would not be linear, since one would have
: ho(kx) = sigma(kx) otimes au(kx) = ksigma(x) otimes k au(x) = k^2 (sigma(x) otimes au(x)) = k^2 ho(x)
for k in K. One can rescue this attempt and restore linearity by imposing additional structure, by defining a map Delta:A ightarrow A imes A, and defining the tensor product representation as : ho = (sigmaotimes au) circ Delta. 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, :x mapsto ho (x) = sigma(x) otimes mbox{Id}_W + mbox{Id}_V otimes au(x) so that the action on the tensor product space is given by : ho(x) (v otimes w) = (sigma(x) v)otimes w + v otimes ( au(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: : ho(xy) = sigma(x) sigma(y) otimes mbox{Id}_W + mbox{Id}_V otimes au(x) au(y). But, in general, this does not equal : ho(x) ho(y) = sigma(x) sigma(y) otimes mbox{Id}_W + sigma(x) otimes au(y) + sigma(y) otimes au(x) + mbox{Id}_V otimes au(x) au(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=
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