General multi-Novikov algebras, multi-differential algebras and their free constructions

TL;DR

This paper develops a general framework for multi-Novikov and multi-differential algebras, extending classical constructions and connecting to Poisson algebras, with explicit free algebra constructions using decorated trees and polynomials.

Contribution

It generalizes Gelfand's construction of Novikov algebras to multi-Novikov and multi-differential contexts, providing new free algebra constructions and establishing connections with Poisson algebras.

Findings

Constructed free noncommuting multi-Novikov algebras from decorated rooted trees.

Extended Gelfand's construction to multi-Novikov and multi-differential algebras.

Linked multi-Novikov algebras with Poisson algebra structures.

Abstract

Motivated by the recent development of noncommutative Novikov algebras and multi-Novikov algebras from the study of regularity structures of stochastic PDEs, this paper gives a general approach to study various multi-Novikov algebras and multi-differential algebras, with close connection with Poisson algebras. The construction of S. Gelfand of Novikov algebras from differential commutative algebras is generalized to this context. Free noncommuting multi-Novikov algebras are constructed from typed decorated rooted trees and from noncommuting multi-differential polynomials with populated conditions.