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 $\oplus_{p=0}^{M}\mathrm{End}S_σ^p(V)$ of graded endomorphisms of the quantum symmetric algebra $S_σ(V)$. 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 $\mathcal{U}_{q}\mathfrak{sl}_{N+1}$ and $σ$ is the action of the $R$-matrix, this trace is a scalar multiple of the quantum trace of type $A$.

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