EXTENSIONS OF THE TENSOR ALGEBRA AND THEIR APPLICATIONS.

This article presents a natural extension of the tensor algebra. In addition to "left multiplications" by vectors, we can consider "derivations" hy co vector s as husic operators on this extended algebra. These two types of operators satisfy an analogue of the canonical commutation relations. This algebra and these operators have the following applications: (i) applications to invariant theory related to tensor products und (ii) applications to immanunts. The latter includes a new method to study the quantum immanants in the universal enveloping algebras of the general linear Lie algebras and their Capclli type Identities ( the higher Capelli identities). [ABSTRACT FROM AUTHOR]