Dual Steenrod algebra

In algebraic topology, through an algebraic operation (dualization), there is an associated commutative algebra[1] 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 $\pi _{*}(MU)$ [2]^{pg 61-62}) with much ease.

Definition

Recall[2]^{pg 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

${\displaystyle \Delta$

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

${\begin{aligned}\deg(\xi _{n})&=2(p^{n}-1)\\\deg(\tau _{n})&=2p^{n}-1\end{aligned}}$

In addition, it has the comultiplication ${\displaystyle \Delta$ defined by

${\begin{aligned}\Delta (\xi _{n})&=\sum _{0\leq i\leq n}\xi _{n-i}^{p^{i}}\otimes \xi _{i}\\\Delta (\tau _{n})&=\tau _{n}\otimes 1+\sum _{0\leq i\leq n}\xi _{n-i}^{p^{i}}\otimes \tau _{i}\end{aligned}}$

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 $\varepsilon$

${\begin{aligned}\eta &:\mathbb {Z} /p\to {\mathcal {A}}_{*}\\\varepsilon &:{\mathcal {A}}_{*}\to \mathbb {Z} /p\end{aligned}}$

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

${\begin{aligned}c(\xi _{0})&=1\\\sum _{0\leq i\leq n}\xi _{n-i}^{p^{i}}c(\xi _{i})&=0\end{aligned}}$

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

Milnor, John (2012-03-29), "The Steenrod algebra and its dual", Topological Library, Series on Knots and Everything, WORLD SCIENTIFIC, vol. 50, pp. 357–382, doi:10.1142/9789814401319_0006, ISBN 978-981-4401-30-2, retrieved 2021-01-05
Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772.

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[1] Milnor, John (2012-03-29), "The Steenrod algebra and its dual", Topological Library, Series on Knots and Everything, WORLD SCIENTIFIC, vol. 50, pp. 357–382, doi:10.1142/9789814401319_0006, ISBN 978-981-4401-30-2, retrieved 2021-01-05

[Ravenel-2] Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772.