Algebra

Lie Algebra, Global Inversion, and the Case for Ontological Two-Sidedness

Authors: Stephen P. Smith
Comments: 18 Pages.

This paper examines whether reciprocal representation provides a mathematically coherent way to express two distinguishable but structurally equivalent sectors without privileging either as primary. The starting point is a pair of isomorphic Lie algebras and their representation within a direct-sum algebra. A natural sector-exchange map is shown to be an involutive Lie-algebra automorphism, while the paired representation forms a Lie subalgebra fixed pointwise by that exchange. The direct sum therefore supplies a precise realization of reciprocal redundancy without requiring nonzero cross-sector brackets. The paper then considers a more general ambient algebra, where closure of the paired representation constrains the cross-sector bracket to be symmetric in its corresponding arguments. The Jacobi identity imposes additional consistency requirements but does not, by itself, require cross-sector brackets to vanish. The limiting case of coincident embeddings is examined, as is an illustrative example in which paired closure coexists with a nonzero cross-sector bracket. The resulting mathematical distinctions are related cautiously to a philosophical proposal of ontological two-sidedness. An appendix further examines how reciprocal exchange may be carried through the curvature construction to an exchange-invariant Yang—Mills Lagrangian, subject to appropriate compatibility conditions. The algebra does not prove that reality has a bilateral ontology; it provides a formal representation in which two distinguishable sides can be related reciprocally without assigning either side ontological priority.
Category: Algebra