Editing Associative algebra (section)

=== 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).