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” 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 the following applications: (i) applications to invariant theory related to tensor products and (ii) applications to immanants. The latter 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).