Fun with k-Algebras

Continuing from last time, suppose we have a bilinear map $f$ from $V \times W$ to $L$ where V, W, and L are vector spaces. Then there is a universal property function $\Phi$ from $V \times W$  to $V \otimes W$ and there is a unique linear map $g$ from $V \otimes W$ to $L$ such that the diagram below commutes:

               $\Phi$

$V \times W$-------> $V \otimes W$

    \                        |

       \                     |

           \                 |

       $f$    \              | $g$

                  \          |

                     \       |

                      _|   \/

                           $L$

The proof is trivial: "f" is used to define a function from the free vector space $F (V \times W)$ to $L$ and then we make a descent by modding by the usual equivalence relation of the tensor product to define the map $g$.

This all looks a bit pedantic, but the point is that any multiplication rule in an algebra $A$ is a bilinear map from $A \times A$ to $A$ and we can now put it in tensor formalism.

In particular consider the algebra $A$ of matrices over a field $k$. Matrix multiplication is associative, and we also have a unit of the algebra: the diagonal matrix with the where the elements are the identity of $k$.  This is a prototypical example of what is called a k-algebra.

Can we formalize the associativity and the unit using the tensor product language? Indeed we can and here is the formal definition:

A k-algebra is a k-vector space $A$ which has a linear map $m : A\otimes A \rightarrow A$ called the multiplication and a unit $u: k \rightarrow A$ such that the following diagrams commute:

                     $m \otimes 1$

$A \otimes A \otimes A$ ----------> $A \otimes A$

              |                             |

              |                             |

  $1 \otimes m$ |                             |  $m$

              |                             |

              |                             |

             \/                            \/

          $A \otimes A$   ---------->    $A$

                           $m$

and

                          $A \otimes A$

                    _                    _

                      |         |         |

$u\otimes 1$       /              |                \ $1\otimes u$

              /                 |                    \

$k \otimes A$                      | $m$               $A \otimes k$

              \                 |                     /

                  \              |                 /

                       _|      \/          |_

                               $A$

Please excuse the sloppiness of the diagrams, it is a real pain to draw them.

So what are those commuting diagrams really saying? 

The first one states that:

$m(m (a\otimes b) \otimes c) = m(a \otimes m(b \otimes c))$

In other words: associativity of the multiplication: (a b) c = a (b c)

The second one defines the algebra unit:

$u(1_k ) a = a u(1_k )$

which means that $u (1_k) = 1_A$

So why do we torture ourselves with this fancy pictorial way of representing trivial properties of algebra? Because now we can do a very powerful thing: reverse the direction of all the arrows. What do get when we do that? We get a brand new concept: the coproduct. Stay tuned next time to explore the wondeful properties of this new mathematical concept.