Heisenberg algebra

The Heisenberg algebra is a 3-dimensional, real Lie algebra with the generators P, Q, R, for which applies

${\displaystyle [P,Q]=R}$

${\displaystyle [P,R]=[Q,R]=0}$

It is the Lie algebra of the Heisenberg group .

presentation

The Heisenberg algebra can be represented as an algebra of matrices by defining

${\displaystyle P={\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\\\end{pmatrix}},\quad Q={\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\\\end{pmatrix}},\quad R={\begin{pmatrix}0&0&1\\0&0&0\\0&0&0\\\end{pmatrix}}}$

and used the commutator of matrices as a Lie bracket . ${\displaystyle [X,Y]=XY-YX}$

generalization

Corresponding to the generalized Heisenberg groups there are also generalized Heisenberg algebras, the Lie algebras of the generalized Heisenberg groups.