A New Factor from E6-Mod to E7-Mod

TL;DR

This paper introduces a novel polynomial representation of the Lie algebra E7, constructs a functor from E6-modules to E7-modules, and explores conditions for irreducibility, with potential applications in quantum field theory and PDEs.

Contribution

It presents a new polynomial representation of E7, a functor from E6-modules to E7-modules, and conditions for irreducibility, extending the understanding of E7 representations.

Findings

New polynomial representation of E7 on 27 variables

Construction of a functor from E6-Modules to E7-Modules

Conditions for irreducibility of the resulting modules

Abstract

We find a new representation of the simple Lie algebra of type $E_7$ on the polynomial algebra in 27 variables, which gives a fractional representation of the corresponding Lie group on 27-dimensional space. Using this representation and Shen's idea of mixed product, we construct a functor from $E_6$-{\bf Mod} to $E_7$-{\bf Mod}. A condition for the functor to map a finite-dimensional irreducible $E_6$-module to an infinite-dimensional irreducible $E_7$-module is obtained. Our general frame also gives a direct polynomial extension from irreducible $E_6$-modules to irreducible $E_7$-modules. The obtained infinite-dimensional irreducible $E_7$-modules are $({\cal G},K)$-modules in terms of Lie group representations. The results could be used in studying the quantum field theory with $E_7$ symmetry and symmetry of partial differential equations.