Quantization of poisson CGL extensions

Abstract

CGL extensions, named after G. Cauchon, K.R. Goodearl, and E.S. 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 $L$. In this thesis, quantum CGL extensions were introduced 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.R. 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 the matrix affine Poisson spaces and more generally that of Schubert cells. In this thesis, an explicit procedure for constructing a symmetric quantum CGL extension from a symmetric integral Poisson CGL extension was described and the uniqueness of such a quantization in a proper sense was established. The quantization procedure readily applies to the examples of symmetric integral Poisson CGL extensions coming from Bott-Samelson varieties.