Lie and pre-Lie theory of Novikov algebras
Ruggero Bandiera, Frédéric Patras
TL;DR
The paper analyzes Novikov algebras through their Lie and pre-Lie structures, linking them to commutative algebras with derivation and exploring enveloping algebras. It develops concrete PBW-type isomorphisms, explicit pre-Lie exponential and logarithm formulas, and a formal flow map, all specialized to the Novikov setting. A central theme is the rich combinatorics relating trees, partitions, Lehmer codes, and tableaux, which yields explicit, computable expressions for enveloping algebras and differential-operator realizations. These results provide a detailed algebraic and combinatorial framework with potential applications to differential calculus, stochastic equations, and numerical methods.
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 Poincaré-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.