Representations of $$E_7$$ E 7

Abstract

Explicit representations of the simple Lie algebra of type \(E_7\) are given . By solving certain partial differential equations, we find the explicit decomposition of the polynomial algebra over the 56-dimensional basic irreducible module of the simple Lie algebra \(E_7\) into a sum of irreducible submodules. Then we study the functor from the module category of \(E_6\) to the module category of \(E_7\) developed. Moreover, we construct a family of irreducible inhomogeneous oscillator representations of the simple Lie algebra of type \(E_7\) on a space of exponential-polynomial functions, related to an explicitly given algebraic variety.