Global and Krull Dimensions of Quantum Weyl Algebras

Given a Hecke symmetry R, A. Giaquinto and J. Zhang defined a natural quantization An(R) of the nth Weyl algebra An based on R and studied many ring theoretic properties of rings A2(Ja, b) (arising from the “Jordan” Hecke symmetry) and An(q, pi, j) (arising from the standard multiparameter Hecke symmetry). Here we compute the global and Krull dimensions in the cases that were left open; namely, we show that over any field k of characteristic zero, gldim(A2(Ja, b)) = Kdim(A2(Ja, b)) = 3 for any a, b k with a ≠ b, and gldim(An( ± 1, pi, j)) = Kdim(An( ± 1, pi, j)) = n.