Gelfand–Kirillov dimension of generalized Weyl algebras (In memory of Guenter Rudolf Krause (1941–2015))

ABSTRACT

Given a generalized Weyl algebra A of degree 1 with the base algebra D, we prove that the difference of the Gelfand–Kirillov dimension of A and that of D could be any positive integer or infinity. Under mild conditions, this difference is exactly 1. As applications, we calculate the Gelfand–Kirillov dimensions of various algebras of interest, including the (quantized) Weyl algebras, ambiskew polynomial rings, noetherian (generalized) down-up algebras, iterated Ore extensions, quantum Heisenberg algebras, universal enveloping algebras of Lie algebras, quantum GWAs, etc.

KEYWORDS:

• Down-up algebra
• Gelfand–Kirillov dimension
• generalized Weyl algebra
• Weyl algebra

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16P90
• 16S36
• 16P40
• 16S32

Acknowledgment

The authors would like to thank the anonymous referee for his/her useful comments and suggestions.