Hypermatrix factors for string and membrane junctions

TL;DR

This paper explores hypermatrix representations derived from Lie algebra decompositions, focusing on their applications to string and membrane junctions, and extends classical group representations to three-vector space products.

Contribution

It introduces hypermatrix factors for classical Lie algebra representations and applies them to model three-junctions of strings and membranes.

Findings

Identifies when three-vector space products appear in Lie algebra decompositions.

Connects Z3-gradings of exceptional Lie algebras to hypermatrix representations.

Provides a framework for formal study of string and membrane junctions.

Abstract

The adjoint representations of the Lie algebras of the classical groups SU(n), SO(n), and Sp(n) are, respectively, tensor, antisymmetric, and symmetric products of two vector spaces, and hence are matrix representations. We consider the analogous products of three vector spaces and study when they appear as summands in Lie algebra decompositions. The Z3-grading of the exceptional Lie algebras provide such summands and provides representations of classical groups on hypermatrices. The main natural application is a formal study of three-junctions of strings and membranes. Generalizations are also considered.