On the Lie transformation algebra of monoids in symmetric monoidal categories

Abstract

We define the Lie transformation algebra of a (not necessarily associative) monoid object $A$ in a $K$-linear symmetric monoidal category $(C,\otimes,1)$, where $K$ is a field. When $A$ is associative and satisfies certain conditions, we describe explicity the Lie transformation algebra and inner derivations of $A$. Additionally, we also show that derivations preserve the nucleus of the monoid $A$.