Cayley Theorems for Loday Algebras

Loday's notoriously elusive "coquecigrues" are meant to relate to Leibniz algebras in the same various ways that groups relate to Lie algebras. However, with the current approaches based on digroups, deadlock has been reached at the analogues of Lie's Third Theorem. Here, adjoint representations appear in the places where regular representations should be expected. The present work, intended as a stimulus to new approaches to the problem, proposes more symmetrical versions of the algebras involved. The fundamental guiding principle is to maintain both left and right actions on a completely equal footing. A coherent and cumulative series of Cayley theorems gives concrete representations of abstract split versions of semigroups, monoids, and groups, based upon the Galois theory of "symmetries of symmetries". Interpreted within monoidal categories, the new group-like objects we present provide a complete left/right split of Hopf algebra structure. The Cayley embedding appears intrinsically as the left/right symmetric part of the coassociativity diagram.