Three main results are obtained in this paper: one generalizes the Atiyah-Bott-Shapiro periodicity equivalence on the category of real Clifford modules, (Theorem 2.2); another establishes the existence of two algebras for real projective representations of the symmetric group Sn and the alternating group An, (Theorem 3.2) and determines their structure, (Theorem 6.1); the third describes all the real projective representations of Sn and An except for some small numbers n, (Theorem 7.2).

Motivated by the problem of giving a functorial (or at least uniform) description of the projective representations for wreath products Gʅ Snin terms of those for G, we study a certain binary operation on the class of “cyclic covering groups with parities”. Along with setting up the basic machinery associated to representations graded by (Ζ/2)ℓ , the main result is a description of the irreducibles for in terms of a (tensorlike) product of those for Aand for B.Finally we describe a programme for producing a PSH-algebra theory in this context, analogous to that of Zelevinsky for the case ℓ=0, and that of the author with with Michael Bean (structure) and with John Humphreys (applications) for the case ℓ=1

A generalization of the definition of combinatorial species is given by considering functors whose domains are categories of finite sets, with various classes of relations as moronisms. Two cases in particular correspond to species for which one has notions of restriction and quotient of structures. Coalgebras and/or Hopf algebras can be associated to such species, the duals of which provide an algebraic framework for studying invariants of structures.

An easy proof will be given to show that for finite dimensional Hopf-algebras with nilpotent augmentation ideal over the field of p elements, the upper and lower Loewy-series coincide. In particular, this holds for the restricted universal envelope of nilpotent Lie-p-algebras with nilpotent p-map.

Let A be an H-simple commutative H-module algebra, with AH = k and dimk H ≤ dimk A < ∞. We show that this implies that A # H is isomorphic to Mn(k), a central simple algebra. We apply this to characterize certain group graded algebras, algebras acted upon by a group as automorphisms, or by a nilpotent Lie algebra as derivations.

In this note we generalise a result of D. Husemoller to certain bipolynomial Hopf algebras and are able to give Hopf algebra decompositions for these. As an easy consequence of our approach we give a simplified derivation of recent results of P. Hoffman on polynomial generators for these algebras; we also give explicit systems of “Borel generators” for a related family of quotient Hopf algebras considered by Hoffman.