AI 中文总结
研究具有特定自同构的顶点代数，通过构造结合代数，建立了不同类型\(V\)模范畴间的同构及等价关系，还针对三个可交换自同构构造双模，证明了特定交织算子空间的同构。
AI 中文摘要
对于具有有限阶自同构\(g\)且满足\(g^T = 1\)（\(T\in\mathbb{N}\)）的顶点代数\(V\)，我们构造了一个结合代数\(\tilde{\mathbf{A}}^{g,\infty}(V)\)，并证明了\(\frac{1}{T}\mathbb{N}\)分次的\(g\)扭曲\(\phi\)协调\(V\)模的范畴与分次\(\tilde{\mathbf{A}}^{g,\infty}(V)\)模的范畴同构。当\(V\)是顶点算子代数时，构造了结合代数\(\mathbf{A}^{g,\infty}(V)\)和\(A^{g,\infty}(V)\)，并建立了可允许\(g\)扭曲\(V\)模和普通\(g\)扭曲\(V\)模的范畴分别与分次\(\mathbf{A}^{g,\infty}(V)\)模和分次\(A^{g,\infty}(V)\)模的范畴同构。通过证明\(\tilde{\mathbf{A}}^{g,\infty}(V)\)与\(\mathbf{A}^{g,\infty}(V)\)同构，得到了\(\frac{1}{T}\mathbb{N}\)分次的\(g\)扭曲\(\phi\)协调\(V\)模的范畴与可允许\(g\)扭曲\(V\)模的范畴之间的等价关系。设\(g_1,g_2,g_3\)是\(V\)的三个可交换有限阶自同构，满足\(g_1g_2 = g_3\)且\(g_i^T = 1\)（\(i = 1,2,3\)），构造了一个\(A^{g_3,\infty}(V)\) - \(A^{g_2,\infty}(V)\)双模\({A}^{g_3,g_2,\infty}(W_1)\)，并证明了\(\binom{W_3}{W_1 \; W_2}\)型交织算子的空间与\(\operatorname{Hom}_{A^{g_3,\infty}(V)}\!\left( {A}^{g_3,g_2,\infty}(W_1) \otimes_{A^{g_2,\infty}(V)} W_2, \, W_3 \right)\)同构。
英文摘要
For a vertex algebra $V$ with a finite-order automorphism $g$ satisfying $g^T = 1$ for some $T \in \mathbb{N}$, we construct an associative algebra $\tilde{\mathbf{A}}^{g,\infty}(V)$ and prove that the category of $\frac{1}{T}\mathbb{N}$-graded $g$-twisted $ϕ$-coordinated $V$-modules is isomorphic to the category of graded $\tilde{\mathbf{A}}^{g,\infty}(V)$-modules. Furthermore, when $V$ is a vertex operator algebra, we construct associative algebras $\mathbf{A}^{g,\infty}(V)$ and $A^{g,\infty}(V)$, and establish that the categories of admissible $g$-twisted $V$-modules and ordinary $g$-twisted $V$-modules are isomorphic to the categories of graded $\mathbf{A}^{g,\infty}(V)$-modules and graded $A^{g,\infty}(V)$-modules, respectively. By proving that $\tilde{\mathbf{A}}^{g,\infty}(V)$ is isomorphic to $\mathbf{A}^{g,\infty}(V)$, we obtain the equivalence between the category of $\frac{1}{T}\mathbb{N}$-graded $g$-twisted $ϕ$-coordinated $V$-modules and the category of admissible $g$-twisted $V$-modules. Let $g_1, g_2, g_3$ be three commuting automorphisms of $V$ of finite order such that $g_1 g_2 = g_3$ and $g_i^T = 1$ for $i = 1, 2, 3$ and some $T \in \mathbb{N}$. Suppose that $W_i$ is a $g_i$-twisted $V$-module for each $i = 1, 2, 3$. We then construct an $A^{g_3,\infty}(V)$-$A^{g_2,\infty}(V)$-bimodule ${A}^{g_3,g_2,\infty}(W_1)$, and prove that the space of intertwining operators of type $\binom{W_3}{W_1 \; W_2}$ is isomorphic to $ \operatorname{Hom}_{A^{g_3,\infty}(V)}\!\left( {A}^{g_3,g_2,\infty}(W_1) \otimes_{A^{g_2,\infty}(V)} W_2, \, W_3 \right). $
Comments51 pages
