# Why Derivations?

This week, I’m going to take a break from the topic that I’ve been writing about for most of this blog because it was a fun excursion into a topic which I found interesting and which seats my research firmly into the bigger picture of Algebra and mathematics in general.

In past months I’ve been focusing on writing down a classification of automorphisms of infinite-dimensional algebras which have a faithful ideal isomorphic to $M_\infty(K)$, the algebra of infinite matrices with only finitely many nonzero elements. Automorphisms of an algebra $A$ can come in two flavors, inner, where the automorphism $\phi: A \rightarrow A$ can be described by conjugation by some invertible element $a \in A$ such that $\phi(x) = a^{-1} x a$ for all $x \in A$. In other words, in the nicest case (where every automorphism of $A$ is inner), we need only look at the set of invertible elements within $A$ to find all the ways that we can “scramble” $A$ without losing any information. An outer automorphism is any automorphism of $A$ which is not inner, so there is some $\phi$ which cannot be described from within $A$.

But this post is not about automorphisms, but rather, derivations. So what are derivations and why are they important? A derivation of an algebra is a linear map $d: A \rightarrow A$ (so a map that doesn’t necessarily preserve the product) which has the property that $d(xy) = x d(y) + d(x) y$ for all $x, y \in A$. The name “derivation” comes from the fact that this is the familiar product rule for derivatives. So given the $K$-algebra of polynomials $K[x]$, the formal derivative is a derivation of the algebra. Just as with automorphisms, derivations come in “inner” and “outer” varieties. Similar to inner automorphisms, an inner derivation is any derivation $d$ that can be described “from within” the algebra $A$, more concretely, there exists some $a \in A$ such that $d(x) = ax - xa$ for all $x \in A$. As above, an outer derivation is any derivation which is not inner. One can check that the derivative is not an inner derivation on $K[x]$ since $\frac{d}{dx} x = 1 \neq 0 = f(x) \cdot x - x \cdot f(x)$ for any polynomial $f(x)$.

So now for the “why.” In my reading about the problem of automorphisms of algebras, I’ve come across many papers which all have the same general outline: 1) Intro, 2) Classification of Automorphisms of $A$, 3) Classification of Derivations of $A$. I was talking with one of my colleagues (an analyst) about this, and he asked me why algebraists were interested in derivations. As an analyst, the derivative operation is a natural tool in his area of study, so derivations make sense for him. But why do algebraists care? The best answer I’ve found is that when you have a classification of derivations and that classification is nice (i.e. all derivations are inner derivations), the algebra with those derivations can be analyzed as a Lie algebra. A Lie algebra is a vector space $V$ over $K$ ( $K$ usually denotes the complex numbers or real numbers) equipped with a bilinear operation $[\cdot, \cdot]: V \times V \rightarrow V$ such that $[x, x] = 0$ and $[x, [y,z]] + [y, [z,x]] + [z, [x,y]] = 0$. Note that $[x,y] = -[y,x]$ by the first property. Specifically, given an algebra, $A$, one can check that the Lie bracket defining $[x,y] = xy - yx$ is a bracket operation which satisfies the previous. A Lie algebra is then a vector space equipped with the “product” given by the bracket operation.

Now one may already see the connections between lie algebras and algebras where every derivation is inner. Given an algebra $A$, you can define or every $a \in A$ $d_a(x) = xa - ax$ and thus collect the set of all inner derivations. If every derivation of $A$ is inner, that means that we may define a Lie bracket on $A$ by $[x,y]$ by $d_x(y)$ and then consider $A$ a Lie algebra. From there, you can delve into the rich theory of Lie algebras to uncover more details about the structure of $A$.

Unfortunately, this paper from 2019 in Journal of Algebra showed that, among other things, the space of outer derivations of the Toeplitz-Jacobson algebra is non-trivial…so I guess we’re talking about inner automorphisms again next post.