** Riv.Mat.Univ.Parma (6) 1 (1998) **

**L. A-M. HANNA**

*A note on the matrix representations of the Lie algebras Lr for quantized Hamiltonians where rs=0*

**Pages:** 149-154

**Received:** 16 February 1998

**Mathematics Subject Classification (2000):** 17B81

**Abstract**:
Consider the Lie algebras *L*^{s}_{r}:
[K_{+},K_{-}]=sK_{0},[K_{0},K_{±}]=
± rK_{±}; r,s in **R**,* K*_{0}
is a Hermitian operator and *K*_{-}= K^{+}_{+}.
In [4], [5] the faithful matrix representations of *L*^{s}_{r}
and ^{c} L^{s}_{r} were discussed for *rs*different
from *0*. In this note we consider the case *rs = 0*. We prove
*L*^{0}_{0} has three types of faithful 3-dimensional
representations, as the least dimension, while *L*^{s}_{0},s
different from *0* and *L*^{0}_{r},r different
from *0*. In this note we consider the case *rs = 0*. We prove
*L*^{0}_{0} has three types of faithful 3-dimensional
representations, as the least dimension, while *L*^{s}_{0},s
different from *0* and *L*^{0}_{r},r different
from *0* have none.

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