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.

Definition

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Recall[2]: 59  that the Steenrod algebra   (also denoted  ) is a graded noncommutative Hopf algebra which is cocommutative, meaning its comultiplication is cocommutative. This implies if we take the dual Hopf algebra, denoted  , or just  , 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

 

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   or odd.

Case of p=2

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In this case, the dual Steenrod algebra is a graded commutative polynomial algebra   where the degree  . Then, the coproduct map is given by

 

sending

 

where  .

General case of p > 2

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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   denote an exterior algebra over   with generators   and  , then the dual Steenrod algebra has the presentation

 

where

 

In addition, it has the comultiplication   defined by

 

where again  .

Rest of Hopf algebra structure in both cases

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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

 

In addition, we will denote   as the kernel of the counit map   which is isomorphic to   in degrees  .

See also

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References

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