Diagrammatics for the smallest quantum coideal and Jones--Wenzl projectors

TL;DR

This paper develops a diagrammatic and algebraic framework for understanding tensor powers of quantum rak{sl}_2's standard representation restricted to a coideal subalgebra, introducing analogues of Jones--Wenzl projectors and -networks.

Contribution

It provides a new diagrammatic and algebraic description of the restriction of quantum rak{sl}_2 tensor powers to a coideal subalgebra, including novel projectors and recursive formulas.

Findings

Describes algebraic and diagrammatic restriction of tensor powers.

Introduces type B/D analogues of Jones--Wenzl projectors.

Provides recursive formulas for -networks.

Abstract

We describe algebraically, diagrammatically and in terms of weight vectors, the restriction of tensor powers of the standard representation of quantum $\mathfrak{sl}_2$ to a coideal subalgebra. We realise the category as module category over the monoidal category of type $\pm1$ representations in terms of string diagrams and via generators and relations. The idempotents projecting onto the quantized eigenspaces are described as type $B/D$ analogues of Jones--Wenzl projectors. As an application we introduce and give recursive formulas for analogues of $\Theta$-networks.