Some Problems in Operad Theory

Abstract

This is a list of some problems and conjectures related to various types of algebras, that is to algebraic operads. Some comments and hints are included.

Associator dependent algebras and Koszul duality

“…Our original motivation came from a slightly more narrow classification problem. In [26], Jean-Louis Loday formulated a question of finding a "small" Koszul operad that acts on the algebra of octonions. That algebra is the only nonassociative normed division algebra over the real numbers (by a theorem of Hurwitz [18], the others, up to isomorphism, are two fields R and C, and the noncommutative algebra of quaternions H).…”

A trick to compute certain Manin products of operads

“…The aim of this paper is to describe a simple method to compute the Manin black products with the operads Ass, Com and preLie. For instance, this allows the computation of Ass • Com and preLie • preLie (Examples 1.5 and 1.7), answering some questions posed by Loday [8] (these had beeen answered already [3,6]). While simple methods to compute preLie • − were already known [2,4], to our knowledge the results are new for the functors Ass • −, Com • −.…”