dual Steenrod algebra

In algebraic topology, through an algebraic operation (dualization), there is an associated commutative algebra{{Citation|last=Milnor|first=John|title=The Steenrod algebra and its dual|date=2012-03-29|url=https://www.worldscientific.com/doi/abs/10.1142/9789814401319_0006|work=Topological Library|volume=50|pages=357–382|series=Series on Knots and Everything|publisher=WORLD SCIENTIFIC|doi=10.1142/9789814401319_0006|isbn=978-981-4401-30-2|access-date=2021-01-05}} 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 $\pi_*(MU)$ {{Cite book|last=Ravenel|first=Douglas C.|url=https://web.math.rochester.edu/people/faculty/doug/mu.html|title=Complex cobordism and stable homotopy groups of spheres|date=1986|publisher=Academic Press|isbn=978-0-08-087440-1|location=Orlando|pages=|oclc=316566772|author-link=Douglas Ravenel}}{{rp|pages=61–62}}) with much ease.

Definition

Recall{{rp|page=59}} 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

References

Category:Algebraic topology

Category:Hopf algebras

Category:Homological algebra