A REMARK ON THE CONJUGATION IN THE STEENROD ALGEBRA

Abstract

Abstract. We investigate the Hopf algebra conjugation, χ, of the mod2 Steenrod algebra, A 2 , in terms of the Hopf algebra conjugation, χ ′ , ofthe mod 2 Leibniz–Hopf algebra. We also investigate the fixed points ofA 2 under χ and their relationship to the invariants under χ ′ . 1. IntroductionThe mod 2 Steenrod algebra A 2 is the free associative graded algebra gen-erated by the Steenrodsquares Sq n [18] of degree n, n ≥ 1, over F 2 subject tothe AdemrelationsSq a Sq b = ⌊ X a/2⌋s=0 b−1−sa−2sSq a+b−s Sq s for 0 < a < 2b.Conventionally, Sq 0 = 1, the multiplicative identity. Topologically, A 2 is thealgebra of stable cohomology operations for ordinary cohomology H ∗ over F 2 .A monomial in A 2 can be written in the form Sq j 1 Sq j 2 ···Sq j r , which weshall denote by Sq j 1 ,j 2 ,...,j r . Admissible monomials form a vector space ba-sis “admissible basis” for A 2 . Milnor [16] determined the graded connectedHopf algebra structure of A 2 by a cocomutative coproduct given by ∆(Sq