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

TL;DR

This paper determines the center of certain quantum algebras associated with Lie algebras and Weyl group elements using quantum cluster algebra techniques, revealing new structural insights and generalizations.

Contribution

It introduces a novel approach to compute centers of quantum algebras via quantum cluster algebra methods, applicable to various algebraic families.

Findings

Center is characterized by the null space of 1+ω.

Generalization of center description to double Schubert Cell algebras.

Centers of additional quadratic algebras are explicitly determined.

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.