The center of ${\mathcal{U}}_q({\mathfrak{n}}_{\omega })$

Abstract

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.

28 pages LaTeX. Relevant references as well as a new section relating to the root-of-unity case have been added. Now in print with minor changes

References in corpus (1)

• Cluster structures on quantum coordinate rings