Derivations in the Steenrod algebra

Question

Let $\mathcal A^\ast$ be the (mod 2) Steenrod algebra.

Question 1: Is there a classification of homogeneous elements $D \in \mathcal A^n$ such that $D^2 = 0$?

Question 2: Is there a classification of homogeneous elements $D \in \mathcal A^n$ such that $D^2 = 0$ and $D(xy) = xD(y) + D(x)y$ whenever $x,y \in H^\ast(X)$ for a space $X$?

Context: I have a cohomology theory which I think is an $H\mathbb F_2$-algebra, and I’m trying to show inductively that the differentials in its AHSS vanish. Since the first nonvanishing differential is a cohomology operation, it seems like such a classification would be useful.

Examples: When $n=1$, there’s only $Sq^1$, which does square to 0. (Not so relevant for the AHSS considerations which start from $n=2$)

When $n=2$, there’s only $Sq^2$, which squares to $Sq^3 Sq^1 \neq 0$.

When $n=3$, we can have linear combinations $aSq^2 Sq^1 + bSq^3$, which (If I’ve done my Adem relations right) squares to $a^2 Sq^5 Sq^1$, which is zero iff $a = 0$. By looking at the cohomology of $\mathbb R \mathbb P^\infty$, I think I’ve convinced myself that $Sq^3$ doesn’t satisfy the above Leibniz rule.

More Context: Specifically, I’m analyzing the AHSS for a 1-periodic multiplicative cohomology theory $R$ whose coefficients are a graded field (I guess if you don’t know that, the above context would seem insufficient, since the differential can be multiplied by coefficients in $R_\ast$, but since $R_\ast$ is a field for me I think this is set up about right.)

Answer 1

The $D$ with $D(xy) = xD(y) + D(x)y$ are the primitives in the Steenrod algebra $A$, which are dual to the indecomposables $\xi_i$ in $A_* = F_2[\xi_i \mid i\ge1]$, so there is one such $D$ in each degree $2^i-1$. With $i=j+1$ these are the Milnor primitives $Q_j$ for $j\ge0$, recursively defined by $Q_0 = Sq^1$ and $Q_j = [Sq^{2^j}, Q_{j-1}]$ for $j\ge1$. Milnor showed that $Q_j^2 = 0$, so the $Q_j$ for $j\ge0$ are the answer to your Question 2. (In degree $n=3$ we have $Q_1 = Sq^3 + Sq^2 Sq^1$, so you might redo that Adem relation calculation.) The $Q_j$ for $j\ge1$ appear as the first nonzero differentials in the AHSS for Morava's $K(j)$.

Answer 2

I have a guess for question 1. Fix $n \geq 0$ and let $E(n)$ be the Hopf subalgebra dual to $\mathbb{F}_2 [\xi_{n+1}, \xi_{n+2}, \dots] / (\xi_i^{2^{n+1}})$. Every $x\in E(n)$ satisfies $x^2=0$, and maybe the converse is true: maybe $x^2=0$ if and only if $x \in E(n)$ for some $n$.