## 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.