Editing Associative algebra (section)

=== 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]].