Heisenberg Lie algebra h2n+1(R)

For an integer $n\geq1$ , the real Heisenberg Lie algebra $\mathfrak h_{2n+1}(\mathbb R)$ is the vector space with basis $x_1,\ldots,x_{n},y_1,\ldots,y_{n},z$ and bilinear alternating bracket determined by

[x_i,y_j]=\delta_{ij}z,\qquad [x_i,x_j]=[y_i,y_j]=[z,x_i]=[z,y_i]=0.

Center and matrix model

Its center and derived algebra are both the line $\mathbb Rz$ . Every nested bracket of length three is zero, so Jacobi is immediate and the algebra is nilpotent of class two. The quotient by its center is abelian of dimension $2n$ .

A faithful matrix model sends $x_i\mapsto E_{1,i+1},\ y_i\mapsto E_{i+1,n+2},\ z\mapsto E_{1,n+2}$ in matrices of size $n+2$ . The subscript records dimension, whereas the catalogue parameter $n$ counts pairs of generators.