Heisenberg Lie algebra h7(C)

The complex Heisenberg Lie algebra $\mathfrak h_{7}(\mathbb C)$ is the vector space with basis $x_1,\ldots,x_{3},y_1,\ldots,y_{3},z$ 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 $\mathbb Cz$ . 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 $6$ .

A faithful matrix model sends $x_i\mapsto E_{1,i+1},\ y_i\mapsto E_{i+1,3+2},\ z\mapsto E_{1,3+2}$ in matrices of size $3+2$ . The subscript records dimension, whereas the catalogue parameter $n$ counts pairs of generators.