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

We propose a general quantum Hamiltonian formalism of a renormalization group (RG) flow with an emphasis on generalized symmetry by interpreting the elementary relationship between homomorphism, quotient ring, and projection. In our formalism, the noninvertible nature of the ideal of a fusion ring realizing the generalized symmetry of an ultraviolet (UV) theory plays a fundamental role in determining condensation rules between anyons, resulting in the infrared (IR) theories. 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.