Lie and pre-Lie theory of Novikov algebras

TL;DR

This paper explores the algebraic and combinatorial structures of Novikov algebras, focusing on their Lie and pre-Lie properties, enveloping algebras, and applications to differential calculus, with specific formulas and statistical phenomena analysis.

Contribution

It provides new formulas and structural insights into Novikov algebras, extending classical and pre-Lie PBW theorems and analyzing their combinatorial and statistical aspects.

Findings

Derived specific formulas for Novikov algebras.

Analyzed the structure of their enveloping algebras.

Investigated statistical phenomena related to trees, tableaux, and permutations.

Abstract

Novikov algebras provide a simple but powerful algebraic axiomatization of important features of classical diferential calculus. We study their structure properties, modeling their relationships with commutative algebras with a derivation, featuring the role of their Lie and pre-Lie structures and analyzing the structure of their enveloping algebras. We focus on the combinatorial analysis of the Poincar\'e-Birkhoff-Witt Theorem (classical and pre-Lie), the pre-Lie exponential and logarithm. The topic is important for applications of the theory and has been treated intensively for pre-Lie algebras. However, specific formulas can be obtained in the Novikov case. We analyze their structure, as well as featuring various remarkable properties. Related statistical phenomena on trees, tableaux and permutations are investigated in this context.