Universal Decomposition Algebras Represent Endomorphisms

The goal of this paper is to supply an explicit description of the universal decomposition algebra of the generic polynomial of degree $n$ into the product of two monic polynomials, one of degree $r$, as a representation of Lie algebras of $n\times n$ matrices with polynomial entries. This is related with the bosonic vertex representation of the Lie algebra $g{l}_{\mathrm{\infty }}$ due to Date, Jimbo, Kashiwara and Miwa.