Knowlpedia

Definition

Heisenberg H_7(R)

Heisenberg H_7(R) as a separately identified real Lie group, with its defining operations and dimension.

the H7(R)H_{7}(\mathbb R) has underlying manifold R3×R3×R\mathbb R^{3}\times \mathbb R^{3}\times \mathbb R and multiplication

(x,y,z)(x′,y′,z′)=(x+x′,y+y′,z+z′+12(x⋅y′−y⋅x′)).(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⋅y=∑jxjyjx\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 77. The inverse is (−x,−y,−z)(-x,-y,-z) and the commutator is (0,0,x⋅y′−y⋅x′)(0,0,x\cdot y^{\prime}-y\cdot x^{\prime}). Thus the center and 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 .

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