Lie pairs

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 ``$⪯$-morphisms'' preserving a surpassing relation $⪯$ that replaces equality. We provide versions of the PBW (Poincare-Birkhoff-Witt) Theorem in these three categories.

40 pages

References in corpus (4)

• Hyperfields for Tropical Geometry I. Hyperfields and dequantization
• Grassman semialgebras and the Cayley-Hamilton theorem
• Linear algebra over T-pairs
• Modules and Lie Semialgebras over Semirings with a Negation Map