Computing the Minimal Model for the Quantum Symmetric Algebra

TL;DR

This paper explores the quantum analogue of a classical theorem relating to the minimal model of the quantum symmetric algebra, using tensor categorial tools to extend representation stability results.

Contribution

It provides a self-contained proof of the quantum analogue of a classical theorem, replacing Olver's lemma with associator information in polynomial GL(infty)-representations.

Findings

Established a quantum analogue of the classical theorem.

Replaced Olver's lemma with associator-based arguments.

Connected classical and quantum symmetric algebra models.

Abstract

In this note, we use some of the tensor categorial machinery developed by the quantum algebra community to study algebraic objects which appear in representation stability. In MR3430359, Sam and Snowden prove that the twisted commutative algebra Sym is Morita equivalent to the horizontal strip category. Their proof relies on a lemma proved by Olver in MR924166. We give a self contained proof that replaces Olver's lemma with information about the associator in the underlying category of polynomial GL(infty)-representations. In fact, we prove a quantum analogue of the theorem. The classical version follows by letting the parameter converge to 1.