Induced structures of operated algebras with applications to multi-Novikov algebras

TL;DR

This paper introduces a unified framework for induced structures of operated algebras using unary-binary operads, and applies it to characterize Novikov and multi-Novikov algebras as induced structures.

Contribution

It formalizes the concept of induced structures in operated algebras via unary-binary operads, unifying previous constructions and providing explicit characterizations.

Findings

Novikov algebra is an induced structure of differential commutative algebra.

Multi-Novikov algebra is an induced structure of multi-differential commutative algebra.

Explicit structure of noncommuting multi-differential commutative algebra is determined.

Abstract

We provide a general notion of induced structures of operated algebras in the context of unary-binary operads. This notion fully captures the binary quadratic relations encoded by a unary-binary operad, thereby unifying and formalizing the various constructions that have appeared in the literature under the informal term of ``induced structures''. As an application, we show that the Novikov algebra and the recently introduced multi-Novikov algebra are the induced structures of the differential commutative algebra and the multi-differential commutative algebra respectively. We also explicitly determine the induced structure of the noncommuting multi-differential commutative algebra.