A New Functor from E₆-mod to E₇-mod

Abstract

We find a new representation of the simple Lie algebra of type E ₇ on the polynomial algebra in 27 variables. Using this representation and Shen's idea of mixed product, we construct a new functor from E ₆-Mod to E ₇-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 framework also gives a direct polynomial extension from irreducible E ₆-modules to irreducible E ₇-modules, which can be used to derive Gel'fand–Zetlin bases for E ₇ from those for E ₆ that can be obtained from those for D ₅ according to our earlier work.

Key Words:

• Functor
• Irreducible module
• Polynomial extension
• Representation
• Simple Lie algebra

2010 Mathematics Subject Classification:

• Primary: 17B10
• 17B25
• Secondary: 17B01

Notes

Communicated by K. Misra.