Recursive Fermion System in Cuntz Algebra.I -- Embeddings of Fermion Algebra into Cuntz Algebra ---

TL;DR

This paper presents recursive methods to embed fermion algebras into Cuntz algebras, generalizing previous constructions and connecting to representations of parafermions, with explicit examples and applications.

Contribution

It introduces recursive constructions for embedding CAR and parafermion algebras into Cuntz algebras, expanding the understanding of their algebraic relationships.

Findings

Explicit embedding of CAR into ${ m U}(1)$-invariant subalgebra of ${ m O}_2$

Generalized embedding of CAR into ${ m O}_{2^p}$

Representation of CAR via permutation and Fock representations

Abstract

Embeddings of the CAR (canonical anticommutation relations) algebra of fermions into the Cuntz algebra ${\cal O}_2$ (or ${\cal O}_{2d}$ more generally) are presented by using recursive constructions. As a typical example, an embedding of CAR onto the U(1)-invariant subalgebra of ${\cal O}_2$ is constructed explicitly. Generalizing this construction to the case of ${\cal O}_{2^p}$, an embedding of CAR onto the U(1)-invariant subalgebra of ${\cal O}_{2^p}$ is obtained. Restricting a permutation representation of the Cuntz algebra, we obtain the Fock representation of CAR. We apply the results to embed the algebra of parafermions of order $p$ into ${\cal O}_{2^p}$ according to the Green's ansatz.