Lie pairs

TL;DR

This paper develops a new theory of Lie semialgebra pairs that generalizes classical Lie algebra concepts using null sets and surpassing relations, with applications to different morphism categories.

Contribution

It introduces Lie pairs with null sets and surpassing relations, extending classical Lie algebra theory and establishing PBW theorems in these new contexts.

Findings

Defined Lie pairs with null sets and surpassing relations

Established PBW theorems for these Lie pairs

Presented examples illustrating the theory

Abstract

Extending the theory of systems, we introduce a theory of Lie semialgebra ``pairs'' which parallels the classical theory of Lie algebras, but with a ``null set'' replacing $0$. A selection of examples is given. These Lie pairs comprise two categories in addition to the universal algebraic definition, one with ``weak Lie morphisms'' preserving null sums, and the other with ``$\preceq$-morphisms'' preserving a surpassing relation $\preceq$ that replaces equality. We provide versions of the PBW (Poincare-Birkhoff-Witt) Theorem in these three categories.