In the $q$-oscillator with commutation relation $aa^+-qa^+a=1$, it is known that the smallest commutator algebra of operators containing the creation and annihilation operators $a^+$ and $a$ is the linear span of $a^+$ and $a$, together with all operators of the form $(a^+)^l[a,a^+]^k$, and $[a,a^+]^ka^l$, where $l$ is a positive integer and $k$ is a nonnegative integer. That is, linear combinations of operators of the form $a^h$ or $(a^+)^h$ with $h\geq 2$ or $h=0$ are outside the commutator algebra generated by $a$ and $a^+$... (read more)

PDF- RINGS AND ALGEBRAS

- QUANTUM ALGEBRA