Bernstein-type inequalities for quantum algebras

We establish Bernstein-type inequalities for the quantum algebras $K_{n,\mathrm{\Gamma }}^{P,Q}(\mathbb{K})$ introduced by K. L. Horton that include the graded quantum Weyl algebra, the quantum symplectic space, the quantum Euclidean space, and quantum Heisenberg algebra etc., obtaining new results and as well as simplified proofs of previously known results. The Krull and global dimensions of certain further localizations of $K_{n,\mathrm{\Gamma }}^{P,Q}(\mathbb{K})$ are computed.