Representations of a non-pointed Hopf algebra

<abstract><p>In this paper, we construct all the indecomposable modules of a class of non-pointed Hopf algebras, which are quotient Hopf algebras of a class of prime Hopf algebras of GK-dimension one. Then the decomposition formulas of the tensor product of any two indecomposable modules are established. Based on these results, the representation ring of the Hopf algebras is characterized by generators and some relations.</p></abstract>

Regulator constants of integral representations of finite groups

Let G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$[G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p$[G]-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of Yakovlev.

Milnor K-theory and the graded representation ring

Let F be a field, let G = Gal(/F) be its absolute Galois group, and let R(G,k) be the representation ring of G over a suitable field k. In this preprint we construct a ring homomorphism from the mod 2 Milnor K-theory k*(F) to the graded ring grR(G,k) associated to Grothendieck's γ-filtration. We study this map in particular cases, as well as a related map involving the W-group of F, rather than G. The latter is an isomorphism in all cases considered.Naturally this echoes the Milnor conjecture (now a theorem), which states that k*(F) is isomorphic to the mod 2 cohomology of the absolute Galois group G, and to the graded Witt ring grW(F).The machinery developed to obtain the above results seems to have independent interest in algebraic topology. We are led to construct an analog of the classical Chern character, which does not involve complex vector bundles and Chern classes but rather real vector bundles and Stiefel-Whitney classes. Thus we show the existence of a ring homomorphism whose source is the graded ring associated to the corresponding K-theory ring KO(X) of the topological space X, again with respect to the γ-filtration, and whose target is a certain subquotient of H*(X, F2).In order to define this subquotient, we introduce a collection of distinguished Steenrod operations. They are related to Stiefel-Whitney classes by combinatorial identities.