On the Vertex Operator Representation of Lie Algebras of Matrices

TL;DR

This paper develops a vertex operator framework for representing Lie algebras of matrices, providing explicit formulas and an alternative description of the bosonic representation of infinite-dimensional Lie algebras.

Contribution

It introduces a $B_r$-valued formal power series encoding the action of elementary endomorphisms, improving existing formulas and offering a new vertex operator perspective.

Findings

Derived a $B_r$-valued structural series for Lie algebra actions.

Improved upon Gatto & Salehyan's formula for generating functions.

Constructed an infinite-dimensional structural series for $gl_ ext{infinity}$.

Abstract

The polynomial ring $B_r:=\mathbb{Q}[e_1,\ldots,e_r]$ in $r$ indeterminates is a representation of the Lie algebra of all the endomorphism of $\mathbb{Q}[X]$ vanishing at powers $X^j$ for all but finitely many $j$. We determine a $B_r$-valued formal power series in $r+2$ indeterminates which encode the images of all the basis elements of $B_r$ under the action of the generating function of elementary endomorphisms of $\mathbb{Q}[X]$, which we call the structural series of the representation. The obtained expression implies (and improves) a formula by Gatto & Salehyan, which only computes, for one chosen basis element, the generating function of its images. For sake of completeness we construct in the last section the $B=B_\infty$-valued structural formal power series which consists in the evaluation of the vertex operator describing the bosonic representation of $gl_{\infty}(\mathbb{Q})$ against the generating function of the standard Schur basis of $B$. This provide an alternative description of the bosonic representation of $gl_{\infty}$ due to Date, Jimbo, Kashiwara and Miwa which does not involve explicitly exponential of differential operators.