Sheared algebra maps and operation bialgebras for mod 2 homology and cohomology

The mod 2 Steenrod algebra $\mathcal{A}$ and Dyer-Lashof algebra $\mathcal{R}$ have both striking similarities and differences arising from their common origins in ``lower-indexed'' algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra $\mathcal{K}$, whose module actions are equivalent to, but quite different from, those of $\mathcal{A}$ and $\mathcal{R}$. The exact relationships emerge as ``sheared algebra bijections'', which also illuminate the role of the cohomology of $\mathcal{K}$. As a bialgebra, $\mathcal{K}^{*}$ has a particularly attractive and potentially useful structure, providing a bridge between those of $\mathcal{A^{*}}$ and $\mathcal{R^{*}}$, and suggesting possible applications to the Miller spectral sequence and the $\mathcal{A}$ structure of Dickson algebras.