Endomorphism algebras and q-traces

For a braided vector space (V,σ) with braiding σ of Hecke type, we introduce three associative algebra structures on the space of graded endomorphisms of the quantum symmetric algebra Sσ(V). We use the second product to construct a new trace. This trace is an algebra morphism with respect to the third product. In particular, when V is the fundamental representation of UqslN+1 and σ is the action of the R-matrix, this trace is a scalar multiple of the quantum trace of type A.