Generalized Yang–Baxter Operators for Dieudonné Modules

Rui Saramago

doi:10.3390/axioms4020177

Abstract

An enrichment of a category of Dieudonné modules is made by considering Yang–Baxter conditions, and these are used to obtain ring and coring operations on the corresponding Hopf algebras. Some examples of these induced structures are discussed, including those relating to the Morava K-theory of Eilenberg–MacLane spaces.

Highlights

Dieudonné modules appear as representations of Hopf algebras, in different settings
This equivalence suggests the definition of categories of Dieudonné modules, which can be enriched by considering universal bilinear products, whose equivalent at the level of Hopf algebras give monoidal structures
We start with categories of Dieudonné modules in their own right, not as the equivalent of categories of Hopf algebras, and enrich them in a different way: we define Yang–Baxter operators on such Dieudonné modules, exploring some examples, and only do we look at the effect these operators might have on the equivalent Hopf algebras

Summary

Introduction

Dieudonné modules appear as representations of Hopf algebras, in different settings. Categories ofHopf algebras are equivalent to those of Dieudonné modules, the equivalence being given by the functor that represents each Hopf algebra by its Dieudonné module. For any Dieudonné module M , the identity map A : M ⊗Z M → M ⊗Z M is trivially a generalized Yang–Baxter operator. We define Dieudonné modules for Hopf algebras in HA∗ .

Results

Conclusion