  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.

Representation Theory;Quantum Algebra