The Heisenberg group is a mathematical structure that arises in the context of group theory and analysis, particularly in the study of nilpotent Lie groups and geometric analysis. It is named after the physicist Werner Heisenberg, although its mathematical development is independent of his work in quantum mechanics. The Heisenberg group can be defined in various contexts, such as algebraically, geometrically, or analytically.
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The real Heisenberg group consists of upper-unitriangular real matrices. In coordinates its multiplication is .