Loosely speaking, groups encode symmetries of (geometric) objects: if we consider a space X with some specific structure (e.g., a Riemannian manifold), a symmetry of X is a bijection f: X→X preserving the natural involved structure (e.g., the Riemannian metric) — here, the compositions and inversions of symmetries yield new symmetries. In a handful of assumptions, the concept of groups captures the essential properties of wide classes of symmetries, and provides powerful tools for related analysis.
KeywordsUnitary Representation Group Homomorphism Isotropy Subgroup Neutral Element Irreducible Unitary Representation
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