Editing Associative algebra (section)

== Representations ==
{{main|Algebra representation}}
A [[representation theory|representation]] of an algebra ''A'' is an algebra homomorphism {{nowrap|''ρ'' : ''A'' → End(''V'')}} from ''A'' to the endomorphism algebra of some vector space (or module) ''V''. The property of ''ρ'' being an algebra homomorphism means that ''ρ'' preserves the multiplicative operation (that is, {{nowrap|1=''ρ''(''xy'') = ''ρ''(''x'')''ρ''(''y'')}} for all ''x'' and ''y'' in ''A''), and that ''ρ'' sends the unit of ''A'' to the unit of End(''V'') (that is, to the identity endomorphism of ''V'').

If ''A'' and ''B'' are two algebras, and {{nowrap|''ρ'' : ''A'' → End(''V'')}} and {{nowrap|''τ'' : ''B'' → End(''W'')}} are two representations, then there is a (canonical) representation {{nowrap|''A'' ⊗ ''B'' → End(''V'' ⊗ ''W'')}} of the tensor product algebra {{nowrap|''A'' ⊗ ''B''}} on the vector space {{nowrap|''V'' ⊗ ''W''}}. However, there is no natural way of defining a [[tensor product]] of two representations of a single associative algebra in such a way that the result is still a representation of that same algebra (not of its tensor product with itself), without somehow imposing additional conditions. Here, by ''[[tensor product of representations]]'',  the usual meaning is intended: the result should be a linear representation of the same algebra 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 {{nowrap|''σ'' : ''A'' → End(''V'')}} and {{nowrap|''τ'' : ''A'' → End(''W'')}}.  One might try to form a tensor product representation {{nowrap|''ρ'' : ''x'' ↦ ''σ''(''x'') ⊗ ''τ''(''x'')}} 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 {{nowrap|''k'' ∈ ''K''}}. One can rescue this attempt and restore linearity by imposing additional structure, by defining an algebra homomorphism {{nowrap|Δ : ''A'' → ''A'' ⊗ ''A''}}, and defining the tensor product representation as
: <math>\rho = (\sigma\otimes \tau) \circ \Delta.</math>
Such a homomorphism Δ is called a [[comultiplication]] if it satisfies certain axioms.  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 co-unital as well. A [[Hopf algebra]] is a bialgebra with an additional piece of structure (the so-called antipode), which allows not only to define the tensor product of two representations, but also the Hom module of two representations (again, similarly to how it is done in the representation theory of groups).

=== Motivation for a Lie algebra ===
{{See also|Lie algebra representation}}
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>.
This shows that this definition of a tensor product is too naive; the obvious fix is to define it such that it is antisymmetric, so that the middle two terms cancel. This leads to the concept of a [[Lie algebra]].