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  "body": "\n## Example\n\nThe (real, three-dimensional) *Heisenberg group* $H$ is the group of real upper unitriangular $3 \\times 3$ matrices\n$$[a, b, c] = \\begin{pmatrix} 1 \u0026 a \u0026 c \\\\ 0 \u0026 1 \u0026 b \\\\ 0 \u0026 0 \u0026 1 \\end{pmatrix}, \\qquad a, b, c \\in \\R,$$\nwith matrix multiplication.\nMultiplying out gives\n$$[a, b, c]\\,[a', b', c'] = [a + a', \\ b + b', \\ c + c' + a b'],$$\nand $[a, b, c]^{-1} = [-a, -b, -c + ab]$.\nSo $H$ is $\\R^3$ with a polynomial group law, a [[Lie group|7JBRGCVJRB]] of dimension $3$.\n\nCheck that $H$ is a [[nilpotent Lie group|DK87YN458D]].\nThe commutator of $g = [a, b, 0]$ and $g' = [a', b', 0]$ is $g g' g^{-1} g'^{-1} = [0, 0, ab' - a'b]$.\nThe [[centre|0DZ93493VN]] of $H$ is $Z = \\set{[0, 0, c] : c \\in \\R}$, and all commutators lie in $Z$.\nSo the lower central series is $H \\supset Z \\supset \\set{1}$, and $H$ is a [[nilpotent group|0Q7MMQHC1P]] of class $2$; it is not abelian.\n\n## Remarks\n\n$H$ is connected and simply connected, being homeomorphic to $\\R^3$.\nIts Lie algebra has a basis $X, Y, Z$ with $[X, Y] = Z$ and $[X, Z] = [Y, Z] = 0$.\nThis is the bracket relation of the position and momentum operators of quantum mechanics, up to a constant, which is the origin of the name.\nMore generally, for $n \\ge 1$ the Heisenberg group $H_{2n+1}$ is $\\R^n \\times \\R^n \\times \\R$ with the law $(x, y, c)(x', y', c') = (x + x', y + y', c + c' + x \\cdot y')$, of dimension $2n + 1$.\nThe matrices $[a, b, c]$ with integer entries form a discrete subgroup $H(\\Z)$, and the quotient $H / H(\\Z)$ is compact.\n\n## Explanations\n\n### L1 human\n\nThe Heisenberg group is the simplest group that is almost, but not quite, commutative.\nTwo of its elements commute up to an extra shift in the third coordinate, and that shift itself commutes with everything.\nIn the matrix picture, multiplying two matrices adds the entries above the diagonal, except that the corner entry picks up an extra product $ab'$.\nThis small failure of commutativity is the algebraic shadow of the uncertainty principle in quantum mechanics.\n",
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