标题： 同态、子结构和理想：重整化群或拓扑序的分层结构的基本但严格的方面 显示英文标题

标题： Homomorphism, substructure and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders

摘要： 我们研究在无质量重整化群（RG）流中连接的共形场论中出现的融合环之间的环同态。 通过解释同态、商环和投影之间的基本关系，我们提出了一种无质量和有质量RG流的一般量子哈密顿量形式，特别强调广义对称性。 在我们的形式中，融合环的理想不可逆性质作为任意子之间的凝聚规则起着基础作用。 我们的代数方法适用于$2+1$维拓扑有序系统中的域墙问题以及对应$1+1$维受隙相的分类，例如。 融合环的理想分解为具有不可逆对称性的受隙相及其对称性破缺（或涌现对称性）模式提供了直接但强有力的约束。 此外，即使在一些特定的无质量RG流连接的同态中，也会出现不太熟悉的同态，我们猜想它们对应于近期文献中的部分可解模型。 我们的工作展示了抽象代数结构，理想，在物理学中的RG中的根本重要性。 显示英文摘要

摘要： We study ring homomorphisms between fusion rings appearing in conformal field theories connected under massless renormalization group (RG) flows. By interpreting the elementary relationship between homomorphism, quotient ring, and projection, we propose a general quantum Hamiltonian formalism of a massless and massive RG flow with an emphasis on generalized symmetry. In our formalism, the noninvertible nature of the ideal of a fusion ring plays a fundamental role as a condensation rule between anyons. Our algebraic method applies to the domain wall problem in $2+1$ dimensional topologically ordered systems and the corresponding classification of $1+1$ dimensional gapped phase, for example. An ideal decomposition of a fusion ring provides a straightforward but strong constraint on the gapped phase with noninvertible symmetry and its symmetry-breaking (or emergent symmetry) patterns. Moreover, even in several specific homomorphisms connected under massless RG flows, less familiar homomorphisms appear, and we conjecture that they correspond to partially solvable models in recent literature. Our work demonstrates the fundamental significance of the abstract algebraic structure, ideal, for the RG in physics.