Modules and Lie Semialgebras over Semirings with a Negation Map

TL;DR

This paper develops the theory of modules and Lie semialgebras over semirings with a negation map, focusing on ELT algebras, and establishes foundational results including a version of Cartan's criterion and a counterexample related to PBW.

Contribution

It introduces the concepts of modules and Lie semialgebras over semirings with a negation map, especially in the ELT algebra context, and proves new theorems including a semisimple criterion and a PBW counterexample.

Findings

ELT version of Cartan's criterion for semisimplicity

Counterexample for naive PBW Theorem in this setting

Development of module and Lie algebra theory over semirings with negation

Abstract

In this article, we present the basic definitions of modules and Lie semialgebras over semirings with a negation map. Our main example of a semiring with a negation map is ELT algebras, and some of the results in this article are formulated and proved only in the ELT theory. When dealing with modules, we focus on linearly independent sets and spanning sets. We define a notion of lifting a module with a negation map, similarly to the tropicalization process, and use it to prove several theorems about semirings with a negation map which possess a lift. In the context of Lie semialgebras over semirings with a negation map, we first give basic definitions, and provide parallel constructions to the classical Lie algebras. We prove an ELT version of Cartan's criterion for semisimplicity, and provide a counterexample for the naive version of the PBW Theorem.