Dual Steenrod algebra

In algebraic topology, through an algebraic operation (dualization), there is an associated commutative algebra from the noncommutative Steenrod algebras called the dual Steenrod algebra. This dual algebra has a number of surprising benefits, such as being commutative and provided technical tools for computing the Adams spectral sequence in many cases (such as $$\pi_*(MU)$$ ) with much ease.

Definition
Recall that the Steenrod algebra $$\mathcal{A}_p^*$$ (also denoted $$\mathcal{A}^*$$) is a graded noncommutative Hopf algebra which is cocommutative, meaning its comultiplication is cocommutative. This implies if we take the dual Hopf algebra, denoted $$\mathcal{A}_{p,*}$$, or just $$\mathcal{A}_*$$, then this gives a graded-commutative algebra which has a noncommutative comultiplication. We can summarize this duality through dualizing a commutative diagram of the Steenrod's Hopf algebra structure: $$\mathcal{A}_p^* \xrightarrow{\psi^*} \mathcal{A}_p^* \otimes \mathcal{A}_p^* \xrightarrow{\phi^*} \mathcal{A}_p^*$$ If we dualize we get maps $$\mathcal{A}_{p,*} \xleftarrow{\psi_*} \mathcal{A}_{p,*} \otimes \mathcal{A}_{p,*}\xleftarrow{\phi_*} \mathcal{A}_{p,*}$$ giving the main structure maps for the dual Hopf algebra. It turns out there's a nice structure theorem for the dual Hopf algebra, separated by whether the prime is $$2$$ or odd.

Case of p=2
In this case, the dual Steenrod algebra is a graded commutative polynomial algebra $$\mathcal{A}_* = \mathbb{Z}/2[\xi_1,\xi_2,\ldots]$$ where the degree $$\deg(\xi_n) = 2^n-1$$. Then, the coproduct map is given by"$\Delta:\mathcal{A}_* \to \mathcal{A}_*\otimes\mathcal{A}_*$"sending"$\Delta\xi_n = \sum_{0 \leq i \leq n} \xi_{n-i}^{2^i}\otimes \xi_i$"where $$\xi_0 = 1$$.

General case of p > 2
For all other prime numbers, the dual Steenrod algebra is slightly more complex and involves a graded-commutative exterior algebra in addition to a graded-commutative polynomial algebra. If we let $$\Lambda(x,y)$$ denote an exterior algebra over $$\mathbb{Z}/p$$ with generators $$x$$ and $$y$$, then the dual Steenrod algebra has the presentation"$\mathcal{A}_* = \mathbb{Z}/p[\xi_1,\xi_2,\ldots]\otimes \Lambda(\tau_0,\tau_1,\ldots)$"where $$\begin{align} \deg(\xi_n) &= 2(p^n - 1) \\ \deg(\tau_n) &= 2p^n - 1 \end{align}$$ In addition, it has the comultiplication $$\Delta:\mathcal{A}_* \to \mathcal{A}_*\otimes\mathcal{A}_*$$ defined by $$\begin{align} \Delta(\xi_n) &= \sum_{0 \leq i \leq n} \xi_{n-i}^{p^i}\otimes \xi_i \\ \Delta(\tau_n) &= \tau_n\otimes 1 + \sum_{0 \leq i \leq n}\xi_{n-i}^{p^i}\otimes \tau_i \end{align}$$ where again $$\xi_0 = 1$$.

Rest of Hopf algebra structure in both cases
The rest of the Hopf algebra structures can be described exactly the same in both cases. There is both a unit map $$\eta$$ and counit map $$\varepsilon$$ $$\begin{align} \eta&: \mathbb{Z}/p \to \mathcal{A}_* \\ \varepsilon&: \mathcal{A}_* \to \mathbb{Z}/p \end{align}$$ which are both isomorphisms in degree $$0$$: these come from the original Steenrod algebra. In addition, there is also a conjugation map $$c: \mathcal{A}_* \to \mathcal{A}_*$$ defined recursively by the equations $$\begin{align} c(\xi_0) &= 1 \\ \sum_{0 \leq i \leq n} \xi_{n-i}^{p^i}c(\xi_i)& = 0

\end{align}$$ In addition, we will denote $$\overline{\mathcal{A}_*}$$ as the kernel of the counit map $$\varepsilon$$ which is isomorphic to $$\mathcal{A}_*$$ in degrees $$> 1$$.