Abstract:Let $q$ be a nonzero complex number that is not a root of unity. In the
$q$-oscillator with commutation relation $ a a^+-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 ${a^+}^l{\left[a,a^+\right]}^k$, and
${\left[a,a^+\right]}^k a ^l$, where $l$ is a nonnegative integer and $k$ is a
positive integer. That is, linear combinations of operator… Show more
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