Bosonic and fermionic representations of endomorphisms of exterior algebras

Abstract

We describe the fermionic and bosonic Fock representations of endomorphisms of the exterior algebra of a Q-vector space of infinite countable dimension. Our main tool is the extension of Schubert derivations, some distinguished kind of Hasse-Schmidt derivations originally defined for exterior algebras only, to the fermionic Fock space.

Schur Polynomials and Plücker degree of Schubert Varieties

“…3.1 Let ∆ λ (exp(t)) be the Schur determinant as in (13), attached to the exponential formal power series. For example…”

“…3 will be shortly proven in Section 5, basing upon the notion of Schubert derivation on an exterior algebra as in [6,9,8] alongwith its extension to an infinite wedge power, as in [10,13].…”

Clifford semialgebras

“…On one hand we intend to walk our first steps towards a representation theory of Lie semialgebras, in the sense of [19,Section 8.3], by giving a closer look to the case of Clifford semialgebras, introduced in Section 5, where we work out the first theoretical basics. Secondly, we put our abstract picture at work in section 7, by providing a proper framework to our Theorem 7.24, which computes, in the same spirit of [2] and [3] in the classical algebra framework, the shape of a generating function that describes the exterior A-semialgebra, introduced and studied in [5], as a representation of endomorphisms of a free A-module. The aim is that of excavating the semialgebraic roots of classical mathematical phenomenologies, to investigate to which extent certain classical results in the theory of representation of certain infinite dimensional Lie algebras can be extended to a tropical context, in which one typically works with modules over semialgebras.…”

“…Our main result is Theorem 7.24, in which we propose a more transparent analog for the "Fermionic version" of the generating function that Date, Jimbo, Kashiwara and Miwa (DJKM) [4] provided to describe the representation of Lie algebras of matrices of infinite size having all but finitely many entries zero. (See [15,Section 5.3] for a concise readable account and [7] and [3] for generalizations.) Additional abundant sources of motivation come from other places in well established literature, showing the potential of our subject to interact with other parts of mathematics.…”