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 [math]\displaystyle{ \pi_*(MU) }[/math][2]pg 61-62) with much ease.
Definition
Recall[2]pg 59 that the Steenrod algebra [math]\displaystyle{ \mathcal{A}_p^* }[/math] (also denoted [math]\displaystyle{ \mathcal{A}^* }[/math]) is a graded noncommutative Hopf algebra which is cocommutative, meaning its comultiplication is cocommutative. This implies if we take the dual Hopf algebra, denoted [math]\displaystyle{ \mathcal{A}_{p,*} }[/math], or just [math]\displaystyle{ \mathcal{A}_* }[/math], 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:
[math]\displaystyle{ \mathcal{A}_p^* \xrightarrow{\psi^*} \mathcal{A}_p^* \otimes \mathcal{A}_p^* \xrightarrow{\phi^*} \mathcal{A}_p^* }[/math]
If we dualize we get maps
[math]\displaystyle{ \mathcal{A}_{p,*} \xleftarrow{\psi_*} \mathcal{A}_{p,*} \otimes \mathcal{A}_{p,*}\xleftarrow{\phi_*} \mathcal{A}_{p,*} }[/math]
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 [math]\displaystyle{ 2 }[/math] or odd.
Case of p=2
In this case, the dual Steenrod algebra is a graded commutative polynomial algebra [math]\displaystyle{ \mathcal{A}_* = \mathbb{Z}/2[\xi_1,\xi_2,\ldots] }[/math] where the degree [math]\displaystyle{ \deg(\xi_n) = 2^n-1 }[/math]. Then, the coproduct map is given by
[math]\displaystyle{ \Delta:\mathcal{A}_* \to \mathcal{A}_*\otimes\mathcal{A}_* }[/math]
sending
[math]\displaystyle{ \Delta\xi_n = \sum_{0 \leq i \leq n} \xi_{n-i}^{2^i}\otimes \xi_i }[/math]
where [math]\displaystyle{ \xi_0 = 1 }[/math].
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 [math]\displaystyle{ \Lambda(x,y) }[/math] denote an exterior algebra over [math]\displaystyle{ \mathbb{Z}/p }[/math] with generators [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math], then the dual Steenrod algebra has the presentation
[math]\displaystyle{ \mathcal{A}_* = \mathbb{Z}/p[\xi_1,\xi_2,\ldots]\otimes \Lambda(\tau_0,\tau_1,\ldots) }[/math]
where
[math]\displaystyle{ \begin{align} \deg(\xi_n) &= 2(p^n - 1) \\ \deg(\tau_n) &= 2p^n - 1 \end{align} }[/math]
In addition, it has the comultiplication [math]\displaystyle{ \Delta:\mathcal{A}_* \to \mathcal{A}_*\otimes\mathcal{A}_* }[/math] defined by
[math]\displaystyle{ \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} }[/math]
where again [math]\displaystyle{ \xi_0 = 1 }[/math].
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 [math]\displaystyle{ \eta }[/math] and counit map [math]\displaystyle{ \varepsilon }[/math]
[math]\displaystyle{ \begin{align} \eta&: \mathbb{Z}/p \to \mathcal{A}_* \\ \varepsilon&: \mathcal{A}_* \to \mathbb{Z}/p \end{align} }[/math]
which are both isomorphisms in degree [math]\displaystyle{ 0 }[/math]: these come from the original Steenrod algebra. In addition, there is also a conjugation map [math]\displaystyle{ c: \mathcal{A}_* \to \mathcal{A}_* }[/math] defined recursively by the equations
[math]\displaystyle{ \begin{align} c(\xi_0) &= 1 \\ \sum_{0 \leq i \leq n} \xi_{n-i}^{p^i}c(\xi_i)& = 0 \end{align} }[/math]
In addition, we will denote [math]\displaystyle{ \overline{\mathcal{A}_*} }[/math] as the kernel of the counit map [math]\displaystyle{ \varepsilon }[/math] which is isomorphic to [math]\displaystyle{ \mathcal{A}_* }[/math] in degrees [math]\displaystyle{ \gt 1 }[/math].
See also
- Adams-Novikov spectral sequence
References
- ↑ Milnor, John (2012-03-29), "The Steenrod algebra and its dual", Topological Library, Series on Knots and Everything (WORLD SCIENTIFIC) 50: pp. 357–382, doi:10.1142/9789814401319_0006, ISBN 978-981-4401-30-2, https://www.worldscientific.com/doi/abs/10.1142/9789814401319_0006, retrieved 2021-01-05
- ↑ 2.0 2.1 Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772. https://web.math.rochester.edu/people/faculty/doug/mu.html.
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