Sets and C^n; Quivers and A-D-E; Triality; Generalized Supersymmetry; and D4-D5-E6

TL;DR

This paper explores the mathematical structures of sets, quivers, and Lie algebras, illustrating their connections to A-D-E diagrams and triality, and discusses their implications for constructing a physics model consistent with experiments.

Contribution

It introduces a novel approach linking set theory, quivers, and Lie algebra representations to fundamental physics models.

Findings

Constructed A-D-E Coxeter-Dynkin diagrams from quivers.

Described triality automorphisms in Spin(8).

Connected Lie algebra structures to physical theories.

Abstract

The relation between Geisteswissenschaft and Naturwissenschaft has been discussed by Munster in hep-th/9305104. The plan of this paper is to begin with the empty set; use it to form sets and quivers (sets of points plus sets of arrows between pairs of points); and then use them to make complex vector spaces and to get the A-D-E Coxeter-Dynkin diagrams. The Dn Spin(2n) Lie algebras have spinor representations to describe fermions. D4 Spin(8) triality gives automorphisms among its vector and two half-spinor representations. D5 Spin(10) contains both Spin(8) and the complexification of the vector representation of Spin(8). E6 contains both Spin(10) and the two half-spinor representations of Spin(10), and therefore contains the adjoint representation of Spin(8) and the complexifications of the vector and the two half-spinor representations of Spin(8). E6 is the basis for construction of a fundamental model of physics that is consistent with experiment (see hep-th/9302030, hep-ph/9301210).