Homology and cohomology operations in terms of differential operators

We consider some actions of the universal Steenrod algebra on the graded algebra of finite Laurent series L ( n ) = 2 [ x 1 ± 1 , … , x n ± 1]compatible with the familiar action of the ordinary Steenrod algebra on H * ((ℝℙ ∞ ) n , 2 ). The induced actions of the lambda algebra and the Dyer–Lashof algebra ℛ on L( n ) − = 2 [ x 1 − 1 , … , x n − 1]are also studied. It turns out that the negative generators of do not act as differential operators on L ( n ), if the Cartan formula holds. We also prove that neither Λ nor ℛ are differential operator algebras when they act non‐trivially on L ( n ) − .