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
[2]: 61–62 ) with much ease.
Recall[2]: 59 that the Steenrod algebra
) is a graded noncommutative Hopf algebra which is cocommutative, meaning its comultiplication is cocommutative.
This implies if we take the dual Hopf algebra, denoted
, 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:
If we dualize we get maps
ψ
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
In this case, the dual Steenrod algebra is a graded commutative polynomial algebra
Then, the coproduct map is given by
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.
denote an exterior algebra over
, then the dual Steenrod algebra has the presentation
In addition, it has the comultiplication
{\displaystyle {\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}}}
The rest of the Hopf algebra structures can be described exactly the same in both cases.
There is both a unit map
and counit map
which are both isomorphisms in degree
: these come from the original Steenrod algebra.
In addition, there is also a conjugation map
defined recursively by the equations
{\displaystyle {\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
as the kernel of the counit map
which is isomorphic to