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)
Recall that the Steenrod algebra
* | |
l{A} | |
p |
l{A}*
l{A}p,*
l{A}*
If we dualize we get maps
* l{A} p
* \xrightarrow{\psi p ⊗
* l{A} p
* \xrightarrow{\phi 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 isl{A}p,*\xleftarrow{\psi*} l{A}p,* ⊗ l{A}p,*\xleftarrow{\phi*} l{A}p,*
2
In this case, the dual Steenrod algebra is a graded commutative polynomial algebra
l{A}*=Z/2[\xi1,\xi2,\ldots]
\deg(\xin)=2n-1
sending\Delta:l{A}*\tol{A}* ⊗ l{A}*
where\Delta\xin=\sum0
2i \xi n-i ⊗ \xii
\xi0=1
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
Λ(x,y)
Z/p
x
y
wherel{A}*=Z/p[\xi1,\xi2,\ldots] ⊗ Λ(\tau0,\tau1,\ldots)
In addition, it has the comultiplication\begin{align} \deg(\xin)&=2(pn-1)\\ \deg(\taun)&=2pn-1 \end{align}
\Delta:l{A}*\tol{A}* ⊗ l{A}*
where again\begin{align} \Delta(\xin)&=\sum0
pi \xi n-i ⊗ \xii\\ \Delta(\taun)&=\taun ⊗ 1+\sum0
pi \xi n-i ⊗ \taui\end{align}
\xi0=1
The rest of the Hopf algebra structures can be described exactly the same in both cases. There is both a unit map
η
\varepsilon
which are both isomorphisms in degree\begin{align} η&:Z/p\tol{A}*\\ \varepsilon&:l{A}*\toZ/p \end{align}
0
c:l{A}*\tol{A}*
In addition, we will denote\begin{align} c(\xi0)&=1\\ \sum0
pi \xi n-i c(\xii)&=0 \end{align}
\overline{l{A}*}
\varepsilon
l{A}*
>1