The odd-primary Kudo–Araki–May algebra of algebraic Steenrod operations and invariant theory

We describe bialgebras of lower-indexed algebraic Steenrod operations over the field with p elements, p an odd prime. These go beyond the operations that can act nontrivially in topology, and their duals are closely related to algebras of polynomial invariants under subgroups of the general linear groups that contain the unipotent upper triangular groups. There are significant differences between these algebras and the analogous one for p = 2, in particular in the nature and consequences of the defining Adem relations.