Endomorphism Algebras and q-Traces

TL;DR

This paper introduces new algebra structures on graded endomorphisms of quantum symmetric algebras for braided vector spaces of Hecke type, leading to a novel trace related to quantum traces in type A.

Contribution

It constructs three associative algebra structures on endomorphisms of quantum symmetric algebras and defines a new trace that generalizes quantum traces for specific representations.

Findings

Defined three algebra structures on graded endomorphisms.

Constructed a new trace as an algebra morphism.

Connected the trace to quantum traces in type A.

Abstract

For a braided vector space $(V,\sigma)$ with braiding $\sigma$ of Hecke type, we introduce three associative algebra structures on the space $\oplus_{p=0}^{M}\mathrm{End}S_\sigma^p(V)$ of graded endomorphisms of the quantum symmetric algebra $S_\sigma(V)$. 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 $\sigma$ is the action of the $R$-matrix, this trace is a scalar multiple of the quantum trace of type $A$.