Operads and bialgebras of multi-indices, and Novikov algebras

TL;DR

This paper develops operadic and bialgebraic structures on noncommutative multi-indices, linking Novikov algebras, rooted trees, and Dyson-Schwinger equations, revealing new algebraic insights and embeddings.

Contribution

It introduces a novel operadic framework for multi-indices, connecting Novikov algebras with bialgebra structures and combinatorial embeddings into the Connes-Kreimer Hopf algebra.

Findings

Established a $bZ$-graded operadic structure on multi-indices.

Embedded the double bialgebra into the Connes-Kreimer Hopf algebra.

Characterized the antipode via a polynomial invariant.

Abstract

Noncommutative multi-indices are noncommutative monomials in a $\mathbb{N}$-indexed family of indeterminates. We define on them a $\mathbb{Z}$-graded operadic structure, with the help of a shifting derivation. Multi-indices of degree 0 are called populated: they form a suboperad, isomorphic to the operad of Novikov algebras. This operadic structure, and the relation between pre-Lie and Novikov algebras, induces two bialgebraic structure in cointeraction on commutative multi-indices. We show how to combinatorially embed this double bialgebra into the Connes-Kreimer Hopf algebra of rooted trees, with its two coproducts based, firstly on cuts, secondly, on contraction of edges, and how this embedding can be characterized by a Dyson-Schwinger equation. We also study the unique polynomial invariant compatible with the two bialgebraic structures on multi-indices and use to describe the antipode for the first coproduct.