The _Heisenberg group_ is a [non-matrix Lie group](Non-matrix%20Lie%20groups.md) sometimes referred to as the _Weyl group_ by physicists. # Representations of the Heisenberg group ## The Schrödinger representation # Heisenberg algebra The _Heisenberg algebra,_ $\mathfrak{h}_{2d+1}$ is a [Lie algebra](Lie%20algebras.md) with $d$ [dimensions](Vector%20space%20dimension.md) is the [real vector space](Real%20vector%20spaces.md) denoted as $\mathbb{R}^{2d+1}=$[$\mathbb{R}^{2d}\oplus\mathbb{R}$](Direct%20sums%20of%20vector%20spaces.md) where its [Lie bracket](Lie%20algebras#^73b451) is $[X_i,Y_j]=\delta_{ij}Z$ where $[X_i,Z]=[Y_i,Z]=0$ and $(i=1...d)$ and $(j=1...d).$ $X_i,Y_j$ and $Z$ form a [basis](linear%20basis) of the [Heisenberg algebra.](Heisenberg%20group.md#Heisenberg%20algebra) %%This is from page 158 and 159 of Woit.%% ## Representations of the Heisenberg algebra #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups/Algebras/LieAlgebras #QuantumMechanics/MathematicalFoundations