Extensions of the tensor algebra and their applications

TL;DR

This paper introduces an extended tensor algebra incorporating derivations by covectors, establishing new operator relations and demonstrating applications in invariant theory and quantum immanants.

Contribution

It presents a novel extension of tensor algebra with derivations, revealing new algebraic structures and applications in invariant theory and quantum algebra.

Findings

Extended tensor algebra with derivations and operator relations.

New methods for studying quantum immanants and Capelli identities.

Applications to invariant theory and tensor product analysis.

Abstract

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).