  CGL extensions, named after G. Cauchon, K. Goodearl, and E. Letzter, are a
special class of noncommutative algebras that are iterated Ore extensions of
associative algebras with compatible torus actions. Examples of CGL extensions
include quantum Schubert cells and quantized coordinate rings of double Bruhat
cells. CGL extensions have been studied extensively in connection with quantum
groups and quantum cluster algebras. For a field $\mathbf{k}$ of characteristic
$0$, let $L=\mathbf{k}[q^{\pm 1}]$ be the $\mathbf{k}$-algebra of Laurent
polynomials in the single variable $q$ and let $\mathbb{K}=\mathbf{k}(q)$ be
the fraction field of $L$. We introduce quantum-CGL extensions as certain
$L$-forms of CGL extensions over $\mathbb{K}$, which have Poisson-CGL
extensions as their semiclassical limits. Poisson-CGL extensions, recently
introduced and systematically studied by K. Goodearl and M. Yakimov, are
certain Poisson polynomial algebras which admit presentations as iterated
Poisson-Ore extensions with compatible torus actions. Examples of Poisson-CGL
extensions include the coordinate rings of matrix affine Poisson spaces and
more generally those of Schubert cells. We describe an explicit procedure for
constructing a symmetric quantum-CGL extension from a symmetric integral
Poisson-CGL extension and establish the uniqueness of such a quantization in a
proper sense.
