In algebraic topology, through an algebraic operation (dualization), there is an associated commutative algebra 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 ) with much ease.
Recall 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:<blockquote></blockquote>If we dualize we get maps<blockquote></blockquote>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.
In this case, the dual Steenrod algebra is a graded commutative polynomial algebra where the degree . Then, the coproduct map is given by<blockquote></blockquote>sending<blockquote></blockquote>where .
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<blockquote></blockquote>where<blockquote></blockquote>In addition, it has the comultiplication defined by<blockquote></blockquote>where again .
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 <blockquote></blockquote>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<blockquote></blockquote>In addition, we will denote as the kernel of the counit map which is isomorphic to in degrees .