We determine the center of a localization of ${mathcal U}_q({mathfrak n}_omega)subseteq {mathcal U}^+_q({mathfrak g})$ by the covariant elements (non-mutable elements) by means of constructions and results from quantum cluster algebras. In our set-up, ${mathfrak g}$ is any finite-dimensional complex Lie algebra and $omega$ is any element in the Weyl group $W$. The non-zero complex parameter $q$ is mostly assumed not to be a root of unity, but our method also gives many details in case $q$ is a primitive root of unity. We point to a new and very useful direction of approach to a general set of problems which we exemplify here by obtaining the result that the center is determined by the null space of $1+omega$. Further, we use this to give a generalization to double Schubert Cell algebras where the center is proved to be given by $omega^{mathfrak a}+omega^{mathfrak c}$. Another family of quadratic algebras is also considered and the centers determined.
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