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 F_p-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 ξ_k 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 ξ_k of degree 2pk - 2 (k≥1) and the exterior algebra in generators τ_k 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).