Super-extensions of tensor algebras and their applications

Abstract: Following arXiv:0909.5586 and arXiv:1411.4125, we construct two super-extensions of the usual tensor algebra through the super-actions of symmetric groups and Hecke algebras respectively. For each extension, we consider a special type of derivations coming from covectors, and study the the space generated, in some special manner, by these derivations and operators from left multiplication by vectors and permutations. Duality theorems of these spaces and the super-actions are proved. As an application, we provide a new proof of the Schur-Sergeev duality theorem, as well as its quantum version.