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... > algebraic topology > manifold > differentiable manifold > Lie group > Heisenberg group

... > topological space > manifold > differentiable manifold > Lie group > Heisenberg group

geometry > differential geometry > differentiable manifold > Lie group > Heisenberg group

topology > differential topology > differentiable manifold > Lie group > Heisenberg group

algebra > group theory > topological group > Lie group > Heisenberg group

geometry > differential geometry > symplectic geometry > Heisenberg group

mathematical physics > quantum field theory > Heisenberg group

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Heisenberg group

In mathematics, the Heisenberg group ${\\displaystyle H}$ , named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form

${\\displaystyle {\egin{pmatrix}1&a&c\\\\0&1&b\\\\0&0&1\\\\\\end{pmatrix}}}$

under the operation of matrix multiplication. Elements a, b and c can be taken from any commutative ring with identity, often taken to be the ring of real numbers (resulting in the "continuous Heisenberg group") or the ring of integers (resulting in the "discrete Heisenberg group").
The continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems, especially in the context of the Stone–von Neumann theorem. More generally, one can consider Heisenberg groups associated to n-dimensional systems, and most generally, to any symplectic vector space.

(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Heisenberg_group)

theta representation

groupe de Heisenberg

francés

URI

http://data.loterre.fr/ark:/67375/PSR-K130R6C6-4

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