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
Famous quotes containing the words algebra, structure and/or basis:
“Poetry has become the higher algebra of metaphors.”
—José Ortega Y Gasset (18831955)
“A committee is organic rather than mechanical in its nature: it is not a structure but a plant. It takes root and grows, it flowers, wilts, and dies, scattering the seed from which other committees will bloom in their turn.”
—C. Northcote Parkinson (19091993)
“The self ... might be regarded as a sort of citadel of the mind, fortified without and containing selected treasures within, while love is an undivided share in the rest of the universe. In a healthy mind each contributes to the growth of the other: what we love intensely or for a long time we are likely to bring within the citadel, and to assert as part of ourself. On the other hand, it is only on the basis of a substantial self that a person is capable of progressive sympathy or love.”
—Charles Horton Cooley (18641929)