  This article presents a natural extension of the tensor algebra. In addition
to "left multiplications" by vectors, we can consider "derivations" by
covectors as basic operators on this extended algebra. These two types of
operators satisfy an analogue of the canonical commutation relations. This
algebra and these operators have some applications: (i) applications to
invariant theory related to tensor products, and (ii) applications to
immanants. The latter one includes a new method to study the quantum immanants
in the universal enveloping algebras of the general linear Lie algebras and
their Capelli type identities (the higher Capelli identities).
