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We present an algebraic framework for interacting extended quantum systems to study complex phenomena characterized by the coexistence and competition of different states of matter. We start by showing how to connect different (spin-particle-gauge) languages by means of exact mappings (isomorphisms) that we name dictionaries and prove a fundamental theorem establishing when two arbitrary languages can be connected. These mappings serve to unravel symmetries which are hidden in one representation but become manifest in another. In addition, we establish a formal link between seemingly unrelated physical phenomena by changing the language of our model description. This link leads to the idea of universality or equivalence. Moreover, we introduce the novel concept of emergent symmetry as another symmetry guiding principle. By introducing the notion of hierarchical languages, we determine …