On the pre-commutative envelopes of commutative algebras

TL;DR

This paper proves that all nilpotent commutative algebras can be embedded into pre-commutative (Zinbiel) algebras via the anti-commutator, highlighting the role of nilpotency in finite dimensions.

Contribution

It establishes the existence of pre-commutative envelopes for nilpotent commutative algebras and clarifies the necessity of nilpotency in finite-dimensional cases.

Findings

Nilpotent commutative algebras can be embedded into Zinbiel algebras.

Nilpotency is necessary for finite-dimensional commutative algebras to have pre-commutative envelopes.

Abstract

We prove that every nilpotent commutative algebra can be embedded into a pre-commutative (Zinbiel) algebra with respect to the anti-commutator operation. For finite-dimensional algebras, the nilpotency condition is necessary for a commutative algebra to have a pre-commutative envelope.