## 6

## Operator algebras

Lie algebras of matrices can be mapped onto Lie algebras of operators in a number of different ways. Three useful matrix algebra to operator algebra mappings are described in this chapter.

### 6.1 Boson operator algebras

It is possible to construct useful operator algebras from Lie algebras. An operator Lie algebra can be constructed from a Lie algebra of *n* × *n* matrices by introducing a set of *n* independent boson creation (*b*_{i}^{†}) and annihilation (*b*_{j}) operators that obey the commutation relations

with all other commutators equal ...