Quantization of Poisson CGL extensions

Abstract

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.