Divided power algebras over an operad

TL;DR

This paper characterizes divided power algebras over reduced operads using polynomial operations, providing two descriptions via symmetric group actions and applying results to level algebras, a class of non-associative commutative algebras.

Contribution

It offers a novel characterization of divided power algebras over operads through polynomial operations and explores their application to level algebras.

Findings

Polynomial operations can be described via invariant elements under symmetric group actions.

The characterization applies to non-associative commutative algebras satisfying the exchange law.

Results extend classical divided power algebra concepts to operadic contexts.

Abstract

The purpose of this paper is to give a characterisation of divided power algebras over a reduced operad. Such a characterisation is given in terms of polynomial operations, following the classical example of divided power algebras. We describe these polynomial operations in two different ways: one way uses invariant elements under the action of the symmetric group, the other coinvariant elements. Our results are then applied to the case of level algebras, which are (non-associative) commutative algebras satisfying the exchange law.