Jones type basic construction on field algebras of $G$-spin models

TL;DR

This paper explores the construction of new field algebras from $G$-spin models using crossed product $C^*$-algebras and quantum doubles, revealing their structure and relation to basic constructions.

Contribution

It introduces a Jones type basic construction for field algebras of $G$-spin models via crossed products with the quantum double $D(G)$, providing explicit algebraic structure.

Findings

Constructed the crossed product $C^*$-algebra as a basic construction.

Showed the iterated crossed product is isomorphic to a basic construction.

Provided concrete structure with order and disorder operators.

Abstract

Let $G$ be a finite group. Starting from the field algebra ${\mathcal{F}}$ of $G$-spin models, one can construct the crossed product $C^*$-algebra ${\mathcal{F}}\rtimes D(G)$ such that it coincides with the $C^*$-basic construction for the field algebra ${\mathcal{F}}$ and the $D(G)$-invariant subalgebra of ${\mathcal{F}}$, where $D(G)$ is the quantum double of $G$. Under the natural $\widehat{D(G)}$-module action on ${\mathcal{F}}\rtimes D(G)$,the iterated crossed product $C^*$-algebra can be obtained, which is $C^*$-isomorphic to the $C^*$-basic construction for ${\mathcal{F}}\rtimes D(G)$ and the field algebra ${\mathcal{F}}$. Furthermore, one can show that the iterated crossed product $C^*$-algebra is a new field algebra and give the concrete structure with the order and disorder operators.