Operads with trivial $\mathbb{A}$-actions

TL;DR

This paper explores operads with trivial actions, establishing an equivalence with pseudo-graded-Perm associative algebras and deriving algebraic properties such as noetherianity and centrality of elements.

Contribution

It introduces a new equivalence between $ ext{A}$-trivial operads and pseudo-graded-Perm associative algebras, revealing their algebraic structure and properties.

Findings

Finitely generated $ ext{A}$-trivial operads are right noetherian.

Such operads have integral Gelfand-Kirillov dimension.

Every element in a prime $ ext{A}$-trivial operad is central.

Abstract

We study operads with trivial $\mathbb{A}$-actions and prove an equivalence between the category of $\mathbb{A}$-trivial operads and that of pseudo-graded-Perm associative algebras. As a consequence, we show that finitely generated $\mathbb{A}$-trivial operads are right noetherian of integral Gelfand-Kirillov dimension and that every element in a prime $\mathbb{A}$-trivial operad is central.