Steenrod Algebra - Hopf Algebra Structure and The Milnor Basis

Hopf Algebra Structure and The Milnor Basis

The Steenrod algebra has more structure than a graded Fp-algebra. It is also a Hopf algebra, so that in particular there is a diagonal or comultiplication map

induced by the Cartan formula for the action of the Steenrod algebra on the cup product. It is easier to describe than the product map, and is given by

The linear dual of ψ makes the (graded) linear dual A* of A into an algebra. Milnor (1958) proved, for p = 2, that A* is a polynomial algebra, with one generator ξk of degree 2k - 1, for every k, and for p>2 the dual Steenrod algebra A* is the tensor product of the polynomial algebra in generators ξk of degree 2pk - 2 (k≥1) and the exterior algebra in generators τk of degree 2pk - 1 (k≥0). The monomial basis for A* then gives another choice of basis for A, called the Milnor basis. The dual to the Steenrod algebra is often more convenient to work with, because the multiplication is (super) commutative. The comultiplication for A* is the dual of the product on A; it is given by

where ξ0=1, and
if p>2

The only primitive elements of A* for p=2 are the, and these are dual to the (the only indecomposables of A).

Read more about this topic:  Steenrod Algebra

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