An algebraic introduction to the Steenrod algebra

TL;DR

This paper introduces the Steenrod algebra through an algebraic approach, avoiding cohomology operations, by representing it as a subalgebra of endomorphisms of a polynomial functor related to vector spaces over Galois fields.

Contribution

It provides an algebraic construction of the Steenrod algebra as a subalgebra of endomorphisms of a polynomial functor, emphasizing its generation by Frobenius map components.

Findings

Steenrod algebra characterized algebraically without cohomology operations

Identifies the algebra as generated by Frobenius map components

Provides a new perspective on the algebra's structure

Abstract

The purpose of these notes is to provide an introduction to the Steenrod algebra in an algebraic manner avoiding any use of cohomology operations. The Steenrod algebra is presented as a subalgebra of the algebra of endomorphisms of a functor. The functor in question assigns to a vector space over a Galois field the algebra of polynomial functions on that vector space: the subalgebra of the endomorphisms of this functor that turns out to be the Steenrod algebra if the ground field is the prime field, is generated by the homogeneous components of a variant of the Frobenius map.