On the Semi-Tensor Product of the Dyer-Lashof and Steenrod Algebras

Abstract

This paper arises out of joint work with F. R. Cohen and F. P. Peterson [5, 2, 3] on the joint structure of infinite loop spaces QX. The homology of such a space is operated on by both the Dyer-Lashof algebra, R, and the opposite of the Steenrod algebra A ∗ . We describe a convenient summary of these actions; let M be the algebra which is R ⊗ A ∗ as a vector space and where multiplication Q 1 ⊗ P J . Q 1’ ⊗ P J’ ∗ is given by applying the Nishida relations in the middle and then the appropriate Adem relations on the ends. Then M is a Hopf algebra which acts on the homology of infinite loop spaces.