Bernstein-type inequalities for quantum algebras

TL;DR

This paper establishes Bernstein-type inequalities for various quantum algebras, providing new results and simplified proofs, and computes dimensions of certain localizations, advancing understanding of quantum algebra structures.

Contribution

It introduces Bernstein-type inequalities for a broad class of quantum algebras and computes dimensions of their localizations, offering new insights and streamlined proofs.

Findings

Established Bernstein-type inequalities for quantum algebras

Computed Krull and global dimensions of localizations

Provided simplified proofs of known results

Abstract

We establish Bernstein-type inequalities for the quantum algebras $K_{n,\Gamma}^{P,Q}(\mathbb{K})$ 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 $K_{n,\Gamma}^{P,Q}(\mathbb{K})$ are computed.