AI 中文总结
该文分类了任意level下parafermion顶点代数$N_k(\boldsymbol{\frak{sl}}_2)$的不可约最高权模，给出Zhu代数的表示，在非整数可允许level及临界level完成分类，补充了正整数level的已有结果。
AI 中文摘要
设$N^k(\boldsymbol{\frak{sl}}_2)$为泛函parafermion顶点代数，$N_k(\boldsymbol{\frak{sl}}_2)$为其单商代数。我们对所有$k\neq0$的不可约最高权$N^k(\boldsymbol{\frak{sl}}_2)$-模进行分类。我们给出Zhu代数$A(N^k(\boldsymbol{\frak{sl}}_2))$的表示：它是四元变量多项式代数的商，由三个显式多项式生成的理想所定义。该表示还表明，$A(N^k(\boldsymbol{\frak{sl}}_2))$是由权为2和3的场的类生成的多项式子代数上秩为3的自由模。不可约最高权$N^k(\boldsymbol{\frak{sl}}_2)$-模由双参数族$L_k[x,y]$参数化，其中$(x,y)\notin\boldsymbol{C}^2$，该族通过自由场实现构造得到。我们还证明，每个$N^k(\boldsymbol{\frak{sl}}_2)$-模$L_k[x,y]$都可作为泛函仿射顶点代数$V^k(\boldsymbol{\frak{sl}}_2)$的不可约权模$M$的$N^k(\boldsymbol{\frak{sl}}_2)$-子模实现。在非整数可允许level下，我们证明$L_k[x,y]$是$N_k(\boldsymbol{\frak{sl}}_2)$-模当且仅当关联的$V^k(\boldsymbol{\frak{sl}}_2)$-模$M$是$L_k(\boldsymbol{\frak{sl}}_2)$-模，这给出了所有非整数可允许level下不可约最高权$N_k(\boldsymbol{\frak{sl}}_2)$-模的分类。我们还对临界level下单parafermion代数$N_{-2}(\boldsymbol{\frak{sl}}_2)$的不可约最高权模进行了分类；而正整数level下不可约$N_k(\boldsymbol{\frak{sl}}_2)$-模此前已由Arakawa、Lam和Yamada完成分类。
英文摘要
Let $N^k(\mathfrak{sl}_2)$ be the universal parafermion vertex algebra and $N_k(\mathfrak{sl}_2)$ its simple quotient. We classify all irreducible highest weight $N^k(\mathfrak{sl}_2)$-modules for every $k\neq 0$. We give a presentation of Zhu's algebra $A(N^k(\mathfrak{sl}_2))$ as a quotient of a polynomial algebra in four variables by an ideal generated by three explicit polynomials. This presentation also shows that $A(N^k(\mathfrak{sl}_2))$ is a free module of rank three over the polynomial subalgebra generated by the classes of the fields of weights two and three. The irreducible highest weight $N^k(\mathfrak{sl}_2)$-modules are parametrized by a two-parameter family $L_k[x,y]$, $(x,y)\in\mathbb C^2$, constructed using a free-field realization. We also prove that each $N^k(\mathfrak{sl}_2)$-module $L_k[x,y]$ can be realized as an $N^k(\mathfrak{sl}_2)$-submodule of an irreducible weight module $M$ for the universal affine vertex algebra $V^k(\mathfrak{sl}_2)$. At non-integral admissible levels, we prove that $L_k[x,y]$ is an $N_k(\mathfrak{sl}_2)$-module if and only if the associated $V^k(\mathfrak{sl}_2)$-module $M$ is an $L_k(\mathfrak{sl}_2)$-module. This gives the classification of irreducible highest weight $N_k(\mathfrak{sl}_2)$-modules at all non-integral admissible levels. We also classify the irreducible highest weight modules for the simple parafermion algebra $N_{-2}(\mathfrak{sl}_2)$ at the critical level. At positive integral levels, the irreducible $N_k(\mathfrak{sl}_2)$-modules were previously classified by Arakawa, Lam, and Yamada.
Comments37 pages
