Hasse--Schmidt Derivations and Cayley--Hamilton Theorem for Exterior Algebras

Abstract

Using the natural notion of {\em Hasse--Schmidt derivations on an exterior algebra}, we relate two classical and seemingly unrelated subjects. The first is the celebrated Cayley--Hamilton theorem of linear algebra, "{\em each endomorphism of a finite-dimensional vector space is a root of its own characteristic polynomial}", and the second concerns the expression of the bosonic vertex operators occurring in the representation theory of the (infinite-dimensional) Heinsenberg algebra.

to appear in Cont. Math. ("Selim Krein Centennial"), 17 pages, Collaboration has been supoorted by the Politecnico di Torino Programme "Finanziamento diffuso della Ricerca", the University of Tel Aviv which hosted the first author, partially supported by INDAM-GNSAGA and "PRIN: Geometria delle varietà algebriche"