:''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)|module]]) which also allows the multiplication of vectors in a [[distributivity|distributive]] and [[associativity|associative]] manner. They are thus special [[algebra over a field|algebras]].
== Definition ==
An associative algebra ''A'' over a [[field (mathematics)|field]] ''K'' is defined to be a vector space over ''K'' together with a ''K''-[[bilinear operator|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 1''x'' = ''x''1 = ''x'' for all ''x'' in ''A'', then we call ''A'' a ''associative algebra with one'' or an '''[[unital]]''' (or '''unitary''') '''associative algebra'''.
Such an algebra is a [[ring (algebra)|ring]], and contains all elements ''a'' of the field ''K'' by identification with ''a''1.
The ''dimension'' of the associative algebra ''A'' over the field ''K'' is its [[Hamel dimension|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 (mathematics)|module]]'', rather than a vector space, over ''K'' with a bilinear form. A unital ''R''-algebra ''A'' can equivalently be defined as a [[ring (algebra)|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'''/''n'''''Z''', the integers [[modular arithmetic|modulo]] ''n'', form a unital associative algebra over '''Z'''/''n'''''Z'''.
See [[algebra (ring theory)]] for more.
== Examples ==
* The square ''n''-by-''n'' [[matrix (mathematics)|matrices]] with entries from the field ''K'' form a unitary associative algebra over ''K''.
* The [[complex number]]s form a 2-dimensional unitary associative algebra over the [[real number]]s.
* 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 [[real matrices (2 x 2)]] form an associative algebra useful in plane mapping.
* The [[polynomial]]s with real coefficients form a unitary associative algebra over the reals.
* Given any [[Banach space]] ''X'', the [[continuous function (topology)|continuous]] [[linear operator]]s ''A'' : ''X'' → ''X'' form a unitary associative algebra (using composition of operators as multiplication); this is in fact a [[Banach algebra]].
* Given any [[topology|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 (mathematics)|limit]] as ''x'' nears infinity is zero.
* The [[Clifford algebra]]s are useful in [[geometry]] and [[physics]].
* [[Incidence algebra]]s of locally finite [[partially ordered set]]s 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 transformation|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 theory|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'':
:<math>M: A \times A \rightarrow A</math>
An associative algebra is an algebra where the map ''M'' has the property
:<math>M \circ (\mbox {Id} \times M) = M \circ (M \times \mbox {Id})</math>
Here, the symbol <math>\circ</math> refers to functional composition, and Id is the identity map: <math>Id(x)=x</math> 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
:<math>( M \circ (\mbox {Id} \times M)) (x,y,z) = M (x, M(y,z))</math>
Similarly, a unital associative algebra can be defined in terms of a unit map
:<math>\eta: K \rightarrow A</math>
which has the property
:<math>M \circ (\mbox {Id} \times \eta ) = s = M \circ (\eta \times \mbox {Id})</math>
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: <math>s:K\times A \rightarrow A</math>; 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 diagram]]s which describe the algebra [[axiom]]s; this defines the structure of a [[coalgebra]].
There is also an abstract notion of [[F-coalgebra]].
== Representations ==
A [[group representation|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 <math>\sigma:A\rightarrow gl(V)</math> and <math>\tau:A\rightarrow gl(W)</math>. One might try to form a tensor product representation <math>\rho: x \mapsto \rho(x) = \sigma(x) \otimes \tau(x)</math> according to how it acts on the product vector space, so that
:<math>\rho(x)(v \otimes w) = (\sigma(x)(v)) \otimes (\tau(x)(w))</math>.
However, such a map would not be linear, since one would have
:<math>\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)</math>
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
:<math>\rho = (\sigma\otimes \tau) \circ \Delta</math>.
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,
:<math>x \mapsto \rho (x) = \sigma(x) \otimes \mbox{Id}_W + \mbox{Id}_V \otimes \tau(x)</math>
so that the action on the tensor product space is given by
:<math>\rho(x) (v \otimes w) = (\sigma(x) v)\otimes w + v \otimes (\tau(x) w) </math>.
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:
:<math>\rho(xy) = \sigma(x) \sigma(y) \otimes \mbox{Id}_W + \mbox{Id}_V \otimes \tau(x) \tau(y)</math>.
But, in general, this does not equal
:<math>\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)</math>.
Equality would hold if the product ''xy'' were antisymmetric (if the product were the [[Lie bracket]], that is, <math>xy \equiv M(x,y) = [x,y]</math>), thus turning the associative algebra into a [[Lie algebra]].
==References==
* Ross Street, ''[http://www-texdev.ics.mq.edu.au/Quantum/Quantum.ps Quantum Groups: an entrée to modern algebra]'' (1998). ''(Provides a good overview of index-free notation)''
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