Bosonic and Fermionic Representations of Endomorphisms of Exterior Algebras

TL;DR

This paper explores bosonic and fermionic Fock space representations of the Lie superalgebra of endomorphisms on an infinite-dimensional exterior algebra, extending Schubert derivations to Fermionic Fock space.

Contribution

It introduces a novel approach to represent the endomorphisms of infinite-dimensional exterior algebras using bosonic and fermionic Fock spaces, leveraging Schubert derivations.

Findings

Extended Schubert derivations to Fermionic Fock space.

Provided explicit fermionic and bosonic Fock representations.

Enhanced understanding of Lie superalgebra representations.

Abstract

We describe the fermionic and bosonic Fock representation of the Lie super-algebra of endomorphisms of the exterior algebra of the ${\mathbb Q}$-vector space of infinite countable dimension, vanishing at all but finitely many basis elements. We achieve the goal by exploiting the extension of the Schubert derivations to the Fermionic Fock space.