 Left-invariant frame of the real Heisenberg group

ID: left-invariant-frame-of-the-real-heisenberg-group

Codex  0 Created 2026-10-05 Updated 2026-10-06

(x, y, z) (a, b, c) = (x + a, y + b, z + c + x b)

, the left-invariant frame is

E_{1} = \partial_{x}, E_{2} = \partial_{y} + x \partial_{z}, E_{3} = \partial_{z} .

[E_{1}, E_{2}] = E_{3}

and the other basis brackets vanish.

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