Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lie pairs

Letterio Gatto, Louis Rowen

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.

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 . 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.
Paper Structure (39 sections, 57 equations)

This paper contains 39 sections, 57 equations.

Table of Contents

  1. Introduction
  2. Shape of the paper
  3. Preliminaries and Notation
  4. Pairs
  5. Substitutes for negation
  6. Pre-negation and negation maps
  7. Weak Property N
  8. Surpassing relations
  9. $\preceq$-morphisms
  10. Identities and varieties
  11. The basic theory of Lie pairs
  12. Lie brackets and Lie pairs
  13. Lie brackets on a free module over a base pair $(\mathcal{C} ,\mathcal{C} _0)$
  14. Negated Lie pairs
  15. Lie pairs with a surpassing relation
  16. ...and 24 more sections

Theorems & Definitions (31)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 21 more