Heisenberg H_2n+1(C)

For every integer $n\ge1$ , the Heisenberg group $H_{2n+1}(\mathbb C)$ has underlying manifold $\mathbb C^{n}\times \mathbb C^{n}\times \mathbb C$ and multiplication

(x,y,z)(x^{\prime},y^{\prime},z^{\prime})=\bigl(x+x^{\prime},y+y^{\prime},z+z^{\prime}+\tfrac12(x\cdot y^{\prime}-y\cdot x^{\prime})\bigr).

The dot product is bilinear $x\cdot y=\sum_jx_jy_j$ (no complex conjugation). The subscript denotes dimension over the displayed field, while the catalogue parameter counts pairs of noncentral coordinates.

Dimensions and structure

This group has real dimension $2(2n+1)$ , complex dimension $2n+1$ . The inverse is $(-x,-y,-z)$ and the commutator is $(0,0,x\cdot y^{\prime}-y\cdot x^{\prime})$ . Thus the center and commutator subgroup are exactly the last coordinate, and the group is two-step nilpotent. Bilinearity of the central term verifies associativity. The underlying affine space makes the group connected and simply connected.

Its complex Lie group structure uses holomorphic multiplication and inversion. Forgetting the complex structure gives the underlying real Lie group; real and holomorphic homomorphisms belong to different categories.

References

Pavel Etingof, Lie Groups and Lie Algebras (MIT lecture notes), §49.1, Example 49.2, p. 266; general-coordinate associativity and commutator verified directly.