Connected quantized Weyl algebras and quantum cluster algebras

We investigate a class of noncommutative algebras, which we call connected quantized Weyl algebras, with a simple description in terms of generators and relations. We already knew of two families, both of which arise from cluster mutation in mutationperiodic quivers, and we show that for generic values of a scalar parameter q these are the only examples. We then investigate the ring-theoretic properties of these two families, determining their prime spectra, automorphism groups and some results on their Krull and global dimensions. The theory of ambiskew polynomial rings and generalised Weyl algebras is useful here and we obtain a description of the height 1 prime ideals in certain generalised Weyl algebras, along with some results on the dimension theory of these rings. We also investigate the semiclassical limit Poisson algebras of the connected quantized Weyl algebras, and compare the prime spectra and Poisson prime spectra of the corresponding rings. We also show that the quantum cluster algebra without coefficients for an acyclic quiver is simple, and extend this result to find a simple localisation in the case where there are coefficients. Finally, we investigate quantum cluster algebra structures related to the connected quantized Weyl algebras discussed earlier, and use these to illustrate the previous result.