Differential Novikov algebras

TL;DR

This work analyzes differential Novikov algebras through operad theory by examining the white and black Manin products of the Novikov operad with itself. It computes the dual operad DerNov^!, proving a direct sum decomposition into a Novikov-like and a bicommutative-like component, each carrying extra identities, and establishes explicit dimension formulas: dim(DerNov^!(1))=1, dim(DerNov^!(2))=2, dim(DerNov^!(3))=12, and for n>=4 dim(DerNov^!(n))=n(n+1)/2+1. It also provides a concrete basis construction for DerNov^!⟨X⟩ by decomposing into two free-algebra types and proving the Dong property for the resulting operad. The results deepen understanding of operad duality in the context of Novikov algebras and their derivations, with precise combinatorial and structural descriptions of the dual objects.

Abstract

In this paper, we consider Novikov algebra with derivation and algebra obtained from its dual operad. It turns out that the obtained dual operad has a connection with bicommutative algebras. The motivation for this work comes from the white and black Manin product of the Novikov operad with itself.

Theorems & Definitions (15)

• Definition 1
• Proposition 1
• Proposition 2
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4