The Heisenberg algebra is a 3-dimensional, real Lie algebra with the generators P, Q, R, for which applies
[
P
,
Q
]
=
R.
{\ displaystyle [P, Q] = R}
[
P
,
R.
]
=
[
Q
,
R.
]
=
0
{\ 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
P
=
(
0
1
0
0
0
0
0
0
0
)
,
Q
=
(
0
0
0
0
0
1
0
0
0
)
,
R.
=
(
0
0
1
0
0
0
0
0
0
)
{\ 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 .
[
X
,
Y
]
=
X
Y
-
Y
X
{\ 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.
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